Let $A$ and $B$ be categories with (arbitrary) products and coproducts and $F : A \rightarrow B$ is an equivalence of categories, then $F$ preserves limits and colimits, hence $F$ preserves arbitrary products and coproducts. Here is my question:
Can we prove that if $F$ is an equivalence, then $F$ preserves (arbitrary) products and coproducts without using limits and colimits?
Thank you!
It may also be nice to know that every equivalence can be promoted to adjoint equivalence. This can be done in a symmetrical fashion, hence both $F$ and $G$ are both left and right adjoints.
This implies the wanted.