TITLE: Confusion over functors regarding universal objects
QUESTION [5 upvotes]: Let $\mathcal C$ be a category and suppose it has all finite products. I want to show that there is a functor $- \times  - \colon \mathcal C \times \mathcal C \to \mathcal C$ which sends $(A,B)$ to the product $A \times B$ (as part of proving that $\mathcal C$ is a monoidal category). However there are possibly lots of choices of products of $A$ and $B$, so I guess the only way this makes sense is for each pair $(A,B)$ is to pick one product structure $A \times B$. However am I allowed to do this with the axiom of choice? Surely it is possible there are a massive amount of objects I have to choose a product structure on (e.g. one for each set, and so having to choose a product structure for every element in a proper class)? 
Obviously this question will generalise to other notions such as choosing a specific tensor product etc. Thanks for any help.

REPLY [4 votes]: I'm not a category expert and these issues use to freak me out, but let's see: according to Mac Lane, a category consists of a set of objects, right?
So, all of your isomorphism classes $A\times B$ (consisting of all objects satisfying the universal property of the produt) cannot have more than a set of objects, right?
Hence, you have a family of sets (all the isomorphisms classes of products $A\times B$, for each pair of objects $A,B$ in your category), and you pick one object of each set.
Isn't that exactly the axiom of choice?