Conservativity of set products on totally supported families (${\prod }_i{f}_i\text{ bijective},A_i,B_i\ne \varnothing ⟹f_i\text{ bijective}$)

If some $f_i$ is not injective, extend two distinct equal-image elements to tuples using common choices at all other coordinates; the product is not injective. If some $f_i$ is not surjective, extend a missing target element to a tuple; the product is not surjective. The axiom of choice supplies these extensions. Thus the set-product functor reflects isomorphisms between families whose coordinates are all nonempty.