# Do I understand product of two categories correctly?

OK, I know what a Cartesian product on sets is. When it comes down to category theory, having a category $A$ and category $B$ is sufficient to construct yet another category $A \times B$. However, underlying details of such an operation remain unclear for me.

All the books I've seen state that $\mathsf{Ob}(A \times B) = \mathsf{Ob}(A) \times \mathsf{Ob}(B)$, which is quite expected. But there are morphism as well, right? So I assume $$\forall a, a' \in A, \forall b, b' \in B:$$ $$\mathsf{Hom}_{A \times B}((a, b), (a', b')) = \mathsf{Hom}_A(a, a') \times \mathsf{Hom}_B(b, b')$$ ... and composition of those to be done in the same Cartesian-product-fashion.

Put in a simpler way, $A \times B$ is just a category formed by all possible combinations of ordered pairs of both 1) objects and 2) morphism between them and nothing else?

• Yes, that is correct. If $f:a\to a'$ in $A$ and $g:b\to b'$ in $B$ then $\langle f,g\rangle:(a,b)\to(a',b')$ in $A\times B$. Feb 27 '18 at 19:40
• What else would you expect to be there? Feb 28 '18 at 9:36