= Conservativity of set products on totally supported families
{title2=$\prod_i f_i\text{ bijective},\quad A_i,B_i\ne\varnothing\Longrightarrow f_i\text{ bijective}$}
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.
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