1. Category and Functor

Category

In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by $A,B,C,X,Y,\cdots ,$ and ‘arrows’ between them denoted by $f,g,\cdots ,$ as in the examplesWe will always have an operation $\circ$ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if $$h=g\circ f{f}_2\circ f_1=g_2\circ g_1.$$

Definition 1.1. A category $\mathcal{C}$ consists of

1.	A class of objects: $\mathrm{Obj}(\mathcal{C})$ (a category is called small if its objects form a set). We will write both $A\in \mathrm{Obj}(\mathcal{C})$ and $A\in \mathcal{C}$ for an object $A$ in $\mathcal{C}$.
2.	A set of morphisms: ${\mathrm{Hom}}_{\mathcal{C}}(A,B)$ for each $A,B\in \mathrm{Obj}(\mathcal{C})$. An element $f\in {\mathrm{Hom}}_{\mathcal{C}}(A,B)$ will be called a morphism from $A$ to $B$, and denoted by$$A\stackrel{f}{\to }B\text{or}f:A\to B.$$ When $\mathcal{C}$ is clear from the context, we will simply write $\mathrm{Hom}(A,B)$ for ${\mathrm{Hom}}_{\mathcal{C}}(A,B)$.
3.	A composition operation $\circ$ between morphisms$$\begin{array}{rl}{\mathrm{Hom}}_{\mathcal{C}}(A,B)\times {\mathrm{Hom}}_{\mathcal{C}}(B,C)& \to {\mathrm{Hom}}_{\mathcal{C}}(A,C),\text{for each}A,B,C\in \mathrm{Obj}(\mathcal{C})\\ f\times g& \to g\circ f,\end{array}$$which will be denoted in terms of a diagram by

These are subject to the following axioms:

1.	Associativity: $h\circ (g\circ f)=(h\circ g)\circ f$ holds, and will be denoted by $h\circ g\circ f$ without ambiguity. This property can be expressed in terms of the following commutative diagram
2.	Identity: for each $A\in \mathrm{Obj}(\mathcal{C})$, there exists $1_A\in {\mathrm{Hom}}_{\mathcal{C}}(A,A)$ called the identity element such that$$f\circ 1_A=f=1_B\circ f,\forall A\stackrel{f}{\to }B,$$i.e. we have the following commutative diagrams

Definition 1.2. A subcategory ${\mathcal{C}}^{\prime }$ of $\mathcal{C}$ (denoted by ${\mathcal{C}}^{\prime }\subset \mathcal{C}$) is a category such that

1.	$\mathrm{Obj}({\mathcal{C}}^{\prime })\subset \mathrm{Obj}(\mathcal{C})$
2.	${\mathrm{Hom}}_{{\mathcal{C}}^{\prime }}(A,B)\subset {\mathrm{Hom}}_{\mathcal{C}}(A,B),\forall A,B\in \mathrm{Obj}({\mathcal{C}}^{\prime })$
3.	compositions in ${\mathcal{C}}^{\prime }$ coincide with that in $\mathcal{C}$ under the above inclusion.

${\mathcal{C}}^{\prime }$ is called a full subcategory of $\mathcal{C}$ if ${\mathrm{Hom}}_{{\mathcal{C}}^{\prime }}(A,B)={\mathrm{Hom}}_{\mathcal{C}}(A,B),\forall A,B\in \mathrm{Obj}({\mathcal{C}}^{\prime })$.

Definition 1.3. A morphism $f:A\to B$ is called an isomorphism (or invertible) if there exists $g:B\to A$ such that $f\circ g=1_B$ and $g\circ f=1_A$, i.e. we have the following commutative diagram Two objects $A,B$ are called isomorphic if there exists an isomorphism $f:A\to B$.

Example 1.4. We will frequently use the following categories.

1.	$\mathcal{C}=\mathsf{Set}$, the category of sets:$$\mathrm{Obj}(\mathcal{C})=\{\text{set}\},{\mathrm{Hom}}_{\mathcal{C}}(A,B)=\{\text{set map}A\to B\}.$$
2.	$\mathcal{C}={\mathsf{Vect}}_k$, the category of vector spaces over a field $k$:$$\mathrm{Obj}(\mathcal{C})=\{\text{k-vector space}\},{\mathrm{Hom}}_{\mathcal{C}}(A,B)=\{\text{k-linear map}A\to B\}.$$${\mathsf{Vect}}_k$ is a subcategory of $\mathsf{Set}$, but not a full subcategory.
3.	$\mathcal{C}=\mathsf{Grp}$, the category of groups:$$\mathrm{Obj}(\mathcal{C})=\{\text{group}\},{\mathrm{Hom}}_{\mathcal{C}}(A,B)=\{\text{group homomorphism}A\to B\}.$$It has a full subcategory $$\mathsf{Ab},\text{the category of abelian groups}.$$
4.	$\mathcal{C}=\mathsf{Ring}$, the category of rings:$$\mathrm{Obj}(\mathcal{C})=\{\text{ring}\},{\mathrm{Hom}}_{\mathcal{C}}(A,B)=\{\text{ring homomorphism}A\to B\}.$$$\mathsf{Ring}$ is a subcategory of $\mathsf{Ab}$, but not a full subcategory. $\mathsf{Ring}$ has a full subcategory $$\mathsf{CRing},\text{the category of commutative rings. }$$

The main object of our interest is$$\boxed{\mathsf{Top}:=\text{the category of topological spaces}}$$

•	whose objects are topological spaces and
•	whose morphisms $f:X\to Y$ are continuous maps.

Example 1.5. Let $\mathcal{C}$ and $\mathcal{D}$ be two categories. We can construct a new category $\mathcal{C}\times \mathcal{D}$, called the product of $\mathcal{C}$ and $\mathcal{D}$, as follows.

•	An object of $\mathcal{C}\times \mathcal{D}$ is a pair $(X,Y)$ of objects $X\in \mathcal{C}$ and $Y\in D$.
•	A morphism $(f,g):(X_1,Y_1)\to (X_2,Y_2)$ is a pair of $f\in {\mathrm{Hom}}_{\mathcal{C}}(X_1,X_2),g\in {\mathrm{Hom}}_{\mathcal{D}}(Y_1,Y_2)$.
•	Compositions are componentwise.

Quotient category and homotopy

Definition 1.6. Let $\mathcal{C}$ be a category. Let $\simeq$ be an equivalence relation defined on each ${\mathrm{Hom}}_{\mathcal{C}}(A,B),A,B\in \mathrm{Obj}(\mathcal{C})$ and compatible with the composition in the following sense $$f_1\simeq f_2,g_1\simeq g_2⟹g_1\circ f_1\simeq g_2\circ f_2.$$The compatibility can be represented by the following diagram We say $\simeq$ defines an equivalence relation on $\mathcal{C}$. The quotient category ${\mathcal{C}}^{\prime }=\mathcal{C}/\simeq$ is defined by

•	$\mathrm{Obj}({\mathcal{C}}^{\prime })=\mathrm{Obj}(\mathcal{C})$
•	${\mathrm{Hom}}_{{\mathcal{C}}^{\prime }}(A,B)={\mathrm{Hom}}_{\mathcal{C}}(A,B)/\simeq ,\forall A,B\in \mathrm{Obj}({\mathcal{C}}^{\prime })$.

Exercise 1.7. Check the definition above is well-defined.

One of the most important equivalence relations in algebraic topology is the homotopy relation.

Let $I=[0,1]$. Let $X\times Y$ denote the topological product of $X,Y\in \mathsf{Top}$.

Definition 1.8. Two morphisms $f_0,f_1:X\to Y$ in $\mathsf{Top}$ are said to be homotopic, denoted by $f_0\simeq f_1$, if$$\exists F:X\times I\to Y\text{such that}F{|}_{X\times \{0\}}=f_0\text{and}F{|}_{X\times \{1\}}=f_1.$$We will also write $F:f_0\simeq f_1$ or $f_0\stackrel{F}{\simeq }{f}_1$ to specify the homotopy $F$. This can be illustrated as

Let $f:X\to Y$ be a morphism in $\mathsf{Top}$. We define its homotopy class$$[f]:=\{g\in \mathrm{Hom}(X,Y)\mid g\simeq f\}.$$We denote$$\begin{array}{c}[X,Y]:=\mathrm{Hom}(X,Y)/\simeq .\end{array}$$

Theorem 1.9. Homotopy defines an equivalence relation on $\mathsf{Top}$.

Proof. We first check that $\simeq$ defines an equivalence relation on morphisms.

•	Reflexivity: Take $F$ such that $F{|}_{X\times t}=f$ for any $t\in I$.
•	Symmetry: Assume we have a homotopy $F:f_0\simeq f_1$. Then reversing $I$ asi.e. taking $\tilde{F}(x,t)=F(x,1-t):X\times I\to Y$, gives $f_1\simeq f_0$ as required.
•	Transitivity: Assume we have two homotopies $F:f_0\simeq f_1$ and $G:f_1\simeq f_2$, then putting them together gives $\tilde{F}:f_0\simeq f_2$ as

We next check $\simeq$ is compatible with compositions.

Let $f_0,f_1:X\to Y$ and $g_0,g_1:Y\to Z$. Assume $f_0\stackrel{F}{\simeq }{f}_1$ and $g_0\stackrel{G}{\simeq }{g}_1$. Then

By transitivity, we have proved the compatibility $g_0\circ f_0\simeq g_0\circ f_1\simeq g_1\circ f_1$.

$\square$

We denote the quotient category of $\mathsf{Top}$ under homotopy relation $\simeq$ by$$h\mathsf{Top}=\mathsf{Top}/\simeq$$with morphisms ${\mathrm{Hom}}_{h\mathsf{Top}}(X,Y)=[X,Y]$.

Definition 1.10. Two topological spaces $X,Y$ are said to have the same homotopy type (or homotopy equivalent) if they are isomorphic in $h\mathsf{Top}$.

Example 1.11. $\mathbb{R}$ and ${\mathbb{R}}^2$ are homotopy equivalent, but not homeomorphic. In other words, they are isomorphic in $h\mathsf{Top}$, but not isomorphic in $\mathsf{Top}$. As we will see, ${\mathbb{R}}^1$ and $S^1$ are not homotopy equivalent.

There is also a relative version of homotopy as follows.

Definition 1.12. Let $A\subset X\in \mathsf{Top}$ and $f_0,f_1:X\to Y$ such that $f_0{|}_A=f_1{|}_A:A\to Y$. We say $f_0$ is homotopic to $f_1$ relative to $A$, denoted by$$f_0\simeq f_1\mathrm{rel}A$$if there exists $F:X\times I\to Y$ such that$$F{|}_{X\times \{0\}}=f_0,F{|}_{X\times \{1\}}=f_1,F{|}_{A\times t}=f_0{|}_A,\forall t\in I.$$

We will also write $F:f_0\simeq f_1\mathrm{rel}A$ or $f_0\stackrel{F}{\simeq }{f}_1\mathrm{rel}A$ to specify the homotopy $F$.

Functor

Definition 1.13. Let $\mathcal{C},\mathcal{D}$ be two categories. A covariant functor (resp. contravariant functor) $F:\mathcal{C}\to \mathcal{D}$ consists of

1.	$F:\mathrm{Obj}(\mathcal{C})\to \mathrm{Obj}(\mathcal{D})$, $A\to F(A)$
2.	${\mathrm{Hom}}_{\mathcal{C}}(A,B)\to {\mathrm{Hom}}_{\mathcal{D}}(F(A),F(B)),\forall A,B\in \mathrm{Obj}(\mathcal{C})$. We denote by$$A\stackrel{f}{\to }B⟹F(A)\stackrel{F(f)}{\to }F(B)$$(resp. ${\mathrm{Hom}}_{\mathcal{C}}(A,B)\to {\mathrm{Hom}}_{\mathcal{D}}(F(B),F(A)),\forall A,B\in \mathrm{Obj}(\mathcal{C})$, denoted by$$A\stackrel{f}{\to }B⟹F(B)\stackrel{F(f)}{\to }F(A).)$$

satisfying

1.	$F(g\circ f)=F(g)\circ F(f)$ (resp. $F(g\circ f)=F(f)\circ F(g)$) for any composable morphisms $f,g$(resp. reversing all arrows in the diagram on the right).
2.	$F(1_A)=1_{F(A)},\forall A\in \mathrm{Obj}(\mathcal{C})$.

$F$ is called faithful (or full) if ${\mathrm{Hom}}_{\mathcal{C}}(A,B)\to {\mathrm{Hom}}_{\mathcal{D}}(F(A),F(B))$ is injective (or surjective) $\forall A,B\in \mathrm{Obj}(\mathcal{C})$. $F$ is called fully faithful if $F$ is both full and faithful.

Example 1.14. The identity functor $1_{\mathcal{C}}:\mathcal{C}\to \mathcal{C}$ maps $$1_{\mathcal{C}}(A)=A,1_{\mathcal{C}}(f)=f$$for any object $A$ and morphism $f$.

Example 1.15. $\forall X\in \mathrm{Obj}(\mathcal{C})$,$$\begin{array}{c}\begin{array}{rcl}\mathrm{Hom}(X,-):\mathcal{C}& \to & \mathsf{Set},\\ A& \mapsto & \mathrm{Hom}(X,A)\end{array}\end{array}$$defines a covariant functor and$$\begin{array}{c}\begin{array}{rcl}\mathrm{Hom}(-,X):\mathcal{C}& \to & \mathsf{Set},\\ A& \mapsto & \mathrm{Hom}(A,X)\end{array}\end{array}$$defines a contravariant functor.

Functors of these two types are called representable (by $X$).

Example 1.16. The forgetful functor $\mathsf{Grp}\to \mathsf{Set}$ (mapping a group to its set of group elements) is representable by the free group with one generator, i.e. $\mathbb{Z}$. We have a set isomorphism $$G={\mathrm{Hom}}_{\mathsf{Grp}}(\mathbb{Z},G).$$

Example 1.17. Let $G$ be an abelian group. Given $X\in \mathsf{Top}$, we will study its n-th cohomology $H^n(X;G)$ with coefficients in $G$. It defines a contravariant functor$$H^n(-;G):h\mathsf{Top}\to \mathsf{Set},X\to H^n(X;G).$$We will see that this functor is representable by the Eilenberg-Maclane space $K(G,n)$ if we work with the subcategory of CW-complexes.

Example 1.18. We define a contravariant functor$$\mathrm{Fun}:\mathsf{Top}\to \mathsf{Ring},X\to \mathrm{Fun}(X)={\mathrm{Hom}}_{\mathsf{Top}}(X,\mathbb{R}).$$$\text{Fun}(X)$ are continuous real functions on $X$. A classical result of Gelfand-Kolmogoroff says that two compact Hausdorff spaces $X,Y$ are homeomorphic (i.e. isomorphic in $\mathsf{Top}$) if and only if $\mathrm{Fun}(X)$ and $\mathrm{Fun}(Y)$ are ring isomorphism (i.e. isomorphic in $\mathsf{Ring}$).

Proposition 1.19. Let $F:\mathcal{C}\to \mathcal{D}$ be a functor. Suppose $f:A\to B$ is an isomorphism in $\mathcal{C}$, then $F(f):F(A)\to F(B)$ is an isomorphism in $\mathcal{D}$.

Proof. Exercise.

$\square$

Natural transformation

Definition 1.20. Let $\mathcal{C},\mathcal{D}$ be two categories and $F,G:\mathcal{C}\to \mathcal{D}$ be two functors. A natural transformation $\tau :F\Rightarrow G$ consists of morphisms$$\tau =\{{\tau }_A:F(A)\to G(A)|\forall A\in \mathrm{Obj}(\mathcal{C})\}$$such that the following diagram commutes for any $A,B\in \mathrm{Obj}(\mathcal{C})$ (here $f:A\to B$ if $F,G$ are covariant and $f:B\to A$ if $F,G$ are contravariant)$\tau$ is called a natural isomorphism if ${\tau }_A$ is an isomorphism for any $A\in \mathrm{Obj}(\mathcal{C})$ and we write $F\simeq G$.

Example 1.21. We consider the following two functors $${\mathrm{GL}}_n,(-{)}^{\times }:\mathsf{CRing}\to \mathsf{Grp}.$$Given a commutative ring $R\in \mathsf{CRing}$, ${\mathrm{GL}}_n(R)$ is the group of invertible $n\times n$ matrices with entries in $R$, and $R^{\times }$ is the multiplicative group of invertible elements of $R$. We can identify $(-{)}^{\times }={\mathrm{GL}}_1$.

The determinant defines a natural transformation $$\det :{\mathrm{GL}}_n\Rightarrow (-{)}^{\times }$$where ${\det }_R:{\mathrm{GL}}_n(R)\to R^{\times }$ is the determinant of the matrix. The naturality of $\det$ is rooted in the fact that the formula for determinant is the same for any coefficient ring. In this way, we can say precisely that taking the determinant of a matrix is a natural operation.

Example 1.22. Let $A,B\in \mathcal{C}$ and $f:A\to B$. We have

•	A natural transformation $$f_{\ast }:\mathrm{Hom}(-,A)\Rightarrow \mathrm{Hom}(-,B)$$for (contravariant) representable functors $\mathrm{Hom}(-,A),\mathrm{Hom}(-,B):\mathcal{C}\to \mathsf{Set}$.
•	A natural transformation $$f^{\ast }:\mathrm{Hom}(B,-)\Rightarrow \mathrm{Hom}(A,-)$$for (covariant) representable functors $\mathrm{Hom}(A,-),\mathrm{Hom}(B,-):\mathcal{C}\to \mathsf{Set}$.

Example 1.23. The above example is a special case of the following construction. Let $A\in \mathcal{C}$.

•	Let $F:\mathcal{C}\to \mathsf{Set}$ be a contravariant functor. Then any $\phi \in F(A)$ induces a natural transformation $$\mathrm{Hom}(-,A)\Rightarrow F$$by assigning $f\in \mathrm{Hom}(B,A)$ to $F(f)(\phi )\in F(B)$.
•	Let $G:\mathcal{C}\to \mathsf{Set}$ be a covariant functor. Then any $\phi \in G(A)$ induces a natural transformation $$\mathrm{Hom}(A,-)\Rightarrow G$$by assigning $f\in \mathrm{Hom}(A,B)$ to $G(f)(\phi )\in G(B)$.

Definition 1.24. Let $F,G,H:\mathcal{C}\to \mathcal{D}$ be functors and ${\tau }_1:F\Rightarrow G,{\tau }_2:G\Rightarrow H$ be two natural transformations. The composition ${\tau }_2\circ {\tau }_1$ is a natural transformation from $F$ to $H$ defined by $$({\tau }_2\circ {\tau }_1{)}_A:F(A)\stackrel{{\tau }_1}{\to }G(A)\stackrel{{\tau }_2}{\to }H(A),\forall A\in \mathrm{Obj}(\mathcal{C}).$$

Definition 1.25. Two categories $\mathcal{C},\mathcal{D}$ are called isomorphic if $\exists F:\mathcal{C}\to \mathcal{D},G:\mathcal{D}\to \mathcal{C}$ such that $F\circ G=1_{\mathcal{D}},G\circ F=1_{\mathcal{C}}$. They are called equivalent if $\exists F:\mathcal{C}\to \mathcal{D},G:\mathcal{D}\to \mathcal{C}$ and two natural isomorphisms $F\circ G\simeq 1_{\mathcal{D}},1_{\mathcal{C}}\simeq G\circ F$. In these cases, we say $F:\mathcal{C}\to \mathcal{D}$ gives an isomorphism/equivalence of categories.

In applications, isomorphism is a too strong condition to impose for most interesting functors. Equivalence is more realistic and equally good essentially. The following proposition is very useful in practice.

Proposition 1.26. Let $F:\mathcal{C}\to \mathcal{D}$ be an equivalence of categories. Then $F$ is fully faithful.

Functor Category

Definition 1.27. Let $\mathcal{C}$ be a small category, and $\mathcal{D}$ be a category. We define the functor category $\mathsf{Fun}(\mathcal{C},\mathcal{D})$

•	Objects: functors from $\mathcal{C}$ to $\mathcal{D}$$$F:\mathcal{C}\to \mathcal{D}.$$
•	Morphisms: natural transformations between two functors (which is indeed a set since $\mathcal{C}$ is small).

The following Yoneda Lemma plays a fundamental role in category theory and applications.

Theorem 1.28 (Yoneda Lemma). Let $\mathcal{C}$ be a category and $A\in \mathcal{C}$. Denote the two functors $$h^A={\mathrm{Hom}}_{\mathcal{C}}(-,A):\mathcal{C}\to \mathsf{Set},h_A={\mathrm{Hom}}_{\mathcal{C}}(A,-):\mathcal{C}\to \mathsf{Set}.$$

1.	Contravariant version: Let $F:\mathcal{C}\to \mathsf{Set}$ be a contravariant functor. Then there is an isomorphism of sets $${\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{C},\mathsf{Set})}\left(h^A,F\right)=F(A).$$This isomorphism is functorial in $A$.
2.	Covariant version: Let $G:\mathcal{C}\to \mathsf{Set}$ be a covariant functor. Then there is an isomorphism of sets $${\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{C},\mathsf{Set})}\left(h_A,G\right)=G(A).$$This isomorphism is functorial in $A$.

The precise meaning of functoriality in $A$ is that we have natural isomorphisms of functors $\mathcal{C}\to \mathsf{Set}$$${\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{C},\mathsf{Set})}\left(h^{(-)},F\right)\simeq F(-),{\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{C},\mathsf{Set})}\left(h_{(-)},G\right)\simeq G(-).$$The required isomorphisms in the above Yoneda Lemma are those maps described in Example 1.23.

One important consequence is that we have set isomorphisms $${\mathrm{Hom}}_{\mathsf{Fun}(\mathcal{C},\mathsf{Set})}\left(h_A,h_B\right)={\mathrm{Hom}}_{\mathcal{C}}(A,B),\forall A,B\in \mathcal{C}$$which are functorial both in $A$ and $B$. This gives rise to a fully faithful functor$$h_{(-)}:\mathcal{C}\to \mathsf{Fun}(\mathcal{C},\mathsf{Set}).$$

Duality

Many concepts and statements in category theory have dual descriptions. It is worthwhile to keep eyes on such dualities. Roughly speaking, the dual of a category-theoretical expression is the result of reversing all the arrows for morphisms, changing each reference to a domain to refer to the target (and vice versa), and reversing the order of composition.

For example, let $\mathcal{C}$ for a category. We can define its opposite category ${\mathcal{C}}^{\text{op}}$ by declaring

•	$\mathrm{Obj}({\mathcal{C}}^{\text{op}})=\mathrm{Obj}(\mathcal{C})$;
•	$f:A\to B$ is a morphism in ${\mathcal{C}}^{\text{op}}$ if and only if $f:B\to A$ is a morphism in $\mathcal{C}$;
•	the composition of two morphisms $g\circ f$ in ${\mathcal{C}}^{\text{op}}$ is the same as the composite $f\circ g$ in $\mathcal{C}$.

A contravariant functor $F:\mathcal{C}\to \mathcal{D}$ is the same as a covariant functor $F:{\mathcal{C}}^{\text{op}}\to \mathcal{D}$. With this help, we can work entirely with covariant functors or contravariant functors. For example, the two statements in Yoneda Lemma are actually the same if we consider opposite categories.

An another example, we will often consider the lifting problem by finding a map $F:X\to E$ such that the following diagram is commutativeThe dual problem is the extension problem by finding a map $G$ such that the dual diagram is commutative

Adjoint functor

Let $\mathcal{C},\mathcal{D}$ be two categories, and let $L:\mathcal{C}\to \mathcal{D},R:\mathcal{D}\to \mathcal{C}$ be two (covariant) functors. The rules$$(A,B)\to {\mathrm{Hom}}_{\mathcal{D}}(L(A),B),(A,B)\to {\mathrm{Hom}}_{\mathcal{C}}(A,R(B)),A\in \mathrm{Obj}(\mathcal{C}),B\in \mathrm{Obj}(\mathcal{D})$$define two functors $${\mathrm{Hom}}_{\mathcal{D}}(L(-),-),{\mathrm{Hom}}_{\mathcal{C}}(-,R(-)):{\mathcal{C}}^{\text{op}}\times \mathcal{D}\to \mathsf{Set}.$$We say $L$ and $R$ are adjoint to one another (more precisely, $L$ is the left adjoint, $R$ is the right adjoint), if there is a natural isomorphism $$\tau :{\mathrm{Hom}}_{\mathcal{D}}(L(-),-)\simeq {\mathrm{Hom}}_{{\mathcal{C}}^{}}(-,R(-));$$that is, for each $A\in \mathrm{Obj}(\mathcal{C}),B\in \mathrm{Obj}(\mathcal{D})$, we have a set isomorphism $${\tau }_{A,B}:{\mathrm{Hom}}_{\mathcal{D}}(L(A),B)={\mathrm{Hom}}_{\mathcal{C}}(A,R(B))$$and this isomorphism is functorial both in $A$ and in $B$. We sometimes write adjoint functors as

Example 1.29 (Free vs Forget). Let $X$ be a set, and $F(X)=\underset{x\in X}{⨁}\mathbb{Z}$ denote the free abelian group generated by $X$. This defines a functor $$\text{Free}:\mathsf{Set}\to \mathsf{Ab},X\to F(X).$$Forgetting the group structure defines a functor (such functor is often called a forgetful functor)$$\text{Forget}:\mathsf{Ab}\to \mathsf{Set},A\to A.$$These two functors are adjoint to each otherIn fact, many "free constructions" in mathematics are left adjoint to certain forgetful functors.

Proposition 1.30. Letbe adjoint functors. Then there are natural transformations $$1_{\mathcal{C}}\Rightarrow R\circ LL\circ R\Rightarrow 1_{\mathcal{D}}.$$

Proof. Given $A\in \mathcal{C}$, the required morphism $A\to RL(A)$ corresponds to the identity $1_{L(A)}:L(A)\to L(A)$ under adjoint. The construction of $L\circ R\Rightarrow 1_{\mathcal{D}}$ is similar.

$\square$

$1_{\mathcal{C}}\Rightarrow R\circ L$ is called the unit of the adjunction. $L\circ R\Rightarrow 1_{\mathcal{D}}$ is called the counit of the adjunction.