TITLE: Definition of the Coproduct of Categories?
QUESTION [6 upvotes]: Let $\{\mathbf{C}_i~|~i \in I\}$ be a family of categories. What is $\coprod_{i\in I}\mathbf{C}_i$ ? Google and my regular sources have failed me. I might be able to anticipate (guess) the definition, but I would prefer not to. The setting I am working in is essentially small additive categories, but I don't think that matters.

REPLY [11 votes]: Intuitions: 

Paste the "diagrams" representing the categories side-by-side;
A book with $I$ pages: page $i$ contains $\mathbf C_i$, the book is $\coprod_{i\in I} \mathbf C_i$.

Formally:

Objects: $\coprod_{i \in I}\operatorname{ob} \mathbf C_i$;
Morphisms: $f: (i, C) \to (i,D)$ for $f: C \to D$ a morphism of $\mathbf C_i$

It is not hard to verify that this yields a category, nor is it hard to verify that it has the universal property of the coproduct in $\mathsf{Cat}$.

As to "guessing": it is a perfectly fine approach to try and guess what some limit/colimit should be. When trying to verify the universal property, you will either succeed, or learn something about why your construction does not work. In either case, you've learned. :)