I was wondering if the product category makes sense for an arbitrary family of (non-necessary small) categories. It is clear that given any finite number of categories, you can construct their product category, even if some of the categories are not small. Furthermore, if $\{\mathcal C_i\}_{i\in I}$ is a family of small categories indexed by a set $I$, you should be able to construct their product $\prod_{i\in I}\mathcal C_i $ which, if I am not wrong, is a subcategory of the functor category $\mathcal C^I$, where $\mathcal C=\bigcup_{i\in I} \mathcal C_i$ (I am not an expert in category theory but according to Wiki $\mathbf{Cat}$ is a Cartesian closed category which means that the exponential of any two objects always exists).
So, my question is: what happen if some of the categories $\mathcal C_i$ is not small (for example if it is $\mathbf{Set}$)? Can we still define the product category of this family?
Remark. As I have already said I'm not an expert in category theory. I'm studying only the basics of the theory as a tool for understanding products, pullbacks and so on. So, As Mac Lane says in his book, my category theory is subordinate to set theory (where I am not an expert either). With this I mean I don't know anything about foundations, so that you should avoid an answer based on this (if my question was related with).
Thanks for your help.
Best Answer
$I$ is a set so there is no problem. You can define in a natural way the category product and your only problem is to verify that
$hom((A_i)_{i\in I} ,(B_j)_{j\in I})=$
$=\{(f_i)_{i\in I}: f_i\in hom(A_i,B_i)\}$
is a set, that is true because $I$ and $hom(A_i,B_i)$ are sets for every $i\in I$