Let $\lbrace \mathcal{C}_i\rbrace$ be a collection of categories, indexed by a set $I$ . The disjoint union $\mathcal{C}$ of these categories is defined as follows:

1. the class of objects of $\mathcal{C}$ is the disjoint union of classes of objects, $\operatorname{Ob}(\mathcal{C}_i)$ , for every $i\in I$ ,
2. the class of morphisms of $\mathcal{C}$ is the disjoint union of classes of morphisms, $\operatorname{Mor}(\mathcal{C}_i)$ , for every $i\in I$ .
3. for objects $A,B$ in $\mathcal{C}$ , if they are objects of $\mathcal{C}_i$ , then $\hom(A,B)$ is the set of morphisms from $A$ to $B$ in $\mathcal{C}_i$ , otherwise, $\hom(A,B):=\varnothing$ .
4. given $\hom(A,B)$ and $\hom(B,C)$ , the composition of morphisms is defined so that, if $A,B,C$ are all objects of some $\mathcal{C}_i$ , the composition is the same as the composition of morphisms defined in $\mathcal{C}_i$ . Otherwise, it is defined as $\varnothing$ .

With the above conditions, one immediately sees that $\mathcal{C}$ is a category, as each $\hom(A,B)$ is a set, associativity of morphism composition and identity morphisms all inherit from the individual categories $\mathcal{C}_i$ .

Remark. If each $\mathcal{C}_i$ is small, so is their disjoint union. In fact, in Cat, the category of small categories, the disjoint union of these categories is their coproduct.