There are occasions when we need to consider objects from different categories being paired up. For example, if $\mathcal{C}$ is a category, then $\hom(A,B)$ where $A,B$ are objects of $\mathcal{C}$ is a set, and we can think of $\hom$ as a functor from $\mathcal{C}^{\operatorname{op}}\times \mathcal{C}$ to the category of sets. But what is $\mathcal{C}^{\operatorname{op}}\times \mathcal{C}$ exactly? We will give this a formal definition presently.

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. Define the Cartesian product $\mathcal{C}\times \mathcal{D}$ of $\mathcal{C}$ and $\mathcal{D}$ as the following pair $(O,M)$ , where

• $O$ is the class consisting of ordered pairs $(X,Y)$ , where $X$ is an object in $\mathcal{C}$ and $Y$ is an object in $\mathcal{D}$
• $M$ is the class consisting of ordered pairs $(f,g)$ , where $f$ is a morphism in $\mathcal{C}$ and $g$ is a morphism in $\mathcal{D}$ .

1. Elements of $O$ are called the objects of $\mathcal{C}\times \mathcal{D}$ and elements of $M$ are called the morphisms of $\mathcal{C}\times\mathcal{D}$ . For each morphism $(f,g)\in M$ , we define the domain and codomain operations $$\operatorname{dom}(f,g):=(\operatorname{dom}(f), \operatorname{dom}(g))\quad\mbox{ and }\quad \operatorname{cod}(f,g):=(\operatorname{cod}(f), \operatorname{cod}(g)).$$ Note that for simplicity, we have used the same symbol $\operatorname{dom}$ and $\operatorname{cod}$ for $\mathcal{C},\mathcal{D}$ , and $\mathcal{C}\times \mathcal{D}$ .
2. Next, for each pair of objects $A,B\in \mathcal{C}\times\mathcal{D}$ , we have a set $\hom(A,B)$ consisting of all morphisms in $\mathcal{C}\times \mathcal{D}$ whose domain is $A$ and codomain is $B$ . Note that $\hom(A,B)$ is a set because it is $\hom(X,Z)\times \hom(Y,T)$ , where $A=(X,Y)$ and $B=(Z,T)$ and each component in the product is assumed to be a set.
3. Finally, for objects $A,B,C$ in $\mathcal{C}\times\mathcal{D}$ , we have a function $\circ$ called composition: $$\circ:\hom(A,B)\times \hom(B,C)\to \hom(A,C).$$ To define $\circ$ , write each object $A,B,C$ as ordered pairs: $A=(X,Y)$ , $B=(Z,T)$ , $C=(U,V)$ . In addition, let $\alpha=(f,g)\in \hom(A,B)$ and $\beta=(p,q)\in \hom(B,C)$ . Then $$\circ(\alpha,\beta):=(\circ_1(f,p),\circ_2(g,q)),$$ where $\circ_1$ and $\circ_2$ are compositions defined in $\mathcal{C}$ and $\mathcal{D}$ respectively, such that $$\circ_1:\hom(X,Z)\times \hom(Z,U)\to \hom(X,U)\mbox{ and }\circ_2:\hom(Y,T)\times \hom(T,V)\to \hom(Y,V).$$ As usual, we write $\beta\circ \alpha$ for $\circ(\alpha,\beta)$ .
4. Now, it is not hard to see that $\mathcal{C}\times \mathcal{D}$ with $\circ$ is a category. For example, let us verify that $(A,B)\ne (C,D)$ implies $\hom(A,B)\cap \hom(C,D)=\varnothing$ . Write $A=(X,Y)$ , $B=(Z,S)$ , $C=(T,U)$ and $D=(V,W)$ . Suppose $\alpha=(f,g)\in \hom(A,B)\cap \hom(C,D)$ . Then $f\in \hom(X,Z)\cap \hom(T,V)$ and $g\in \hom(Y,S)\cap \hom(U,W)$ . But this implies $X=T$ , $Z=V$ , $Y=U$ , and $S=W$ . So $A=(X,Y)=(T,U)=C$ and $B=(Z,S)=(V,W)=D$ .

Remarks.

• The above construction can be generalized to $n$ -fold Cartesian products. If $\mathcal{C}_1,\ldots,\mathcal{C}_n$ be categories. Then $\mathcal{C}:=\mathcal{C}_1\times\cdots \mathcal{C}_n$ can be defined much the same way as in the case $n=2$ . $\mathcal{C}$ is a category and is sometimes written $\prod_{i=1}^n \mathcal{C}_i$ .
• Associated with this product, we can form $n$ (covariant) functors called projection functors $\Pi_i:\mathcal{C}\to \mathcal{C}_i$ , given by $\Pi_i(A)=A_i$ and $\Pi_i(\alpha)=\alpha_i$ , where $A=(A_1,\ldots,A_n)$ and $\alpha=(\alpha_1,\ldots,\alpha_n)$ .
• The product $\mathcal{C}$ of $\mathcal{C}_i$ also enjoys the universal property that for every category $\mathcal{D}$ and functors $F_i: \mathcal{D}\to \mathcal{C}_i$ , there is a unique functor $F:\mathcal{D}\to \mathcal{C}$ such that $\Pi_i\circ F=F_i$ (in other words, $F_i$ factors through $F$ ).
• If fact, any category that enjoys the universal property described above is naturally equivalent to the product of $\mathcal{C}_i$ . We may actually define product category this way, and then prove its existence using the construction that is given as the definition at the beginning of this article.
• More generally, we can define arbitrary (direct) product of categories. The definition is completely similar to the one above. If $\lbrace \mathcal{C}_i\mid i\in I\rbrace$ is a family of categories indexed by a set $I$ , we often write $\prod_{i\in I} \mathcal{C}_i$ as the product category. Objects and morphisms are written $(A_i)_{i\in I}$ and $(\alpha_i)_{i\in I}$ respectively. When all the $\mathcal{C}_i$ are identical, say, equal to $\mathcal{C}$ , we also write the product as $\mathcal{C}^I$ , and call it the $I$ -fold direct product of $\mathcal{C}$ .
• The existence of the product of categories indexed by an arbitrary set shows that the category of (small) categories Cat has products.
• Let $\mathcal{C}=\mathcal{D}\times\mathcal{E}$ . Then we may identify $\mathcal{D}$ as a subcategory of $\mathcal{C}$ : for each object $E$ in $\mathcal{E}$ , define $F_E:\mathcal{D}\to\mathcal{C}$ , by $F_E(A):=(A,E)$ and $F(\alpha)=(\alpha,1_E)$ . Then $F_E$ is a faithful functor. The image $\mathcal{C}_E$ of $F_E$ (with objects $(A,E)$ and morphisms $(\alpha,1_E)$ ) is a subcategory of $\mathcal{C}$ . It is not hard to see that $\mathcal{D}$ and $\mathcal{C}_E$ are isomorphic as categories.
• The above also shows that for any objects $A,B$ in $\mathcal{E}$ , $\mathcal{C}_A$ and $\mathcal{C}_B$ are isomorphic.