Does the existence of products in the category of sets imply the Axiom of Choice?

Question

If for every family $(X_i)_{i\in I}$ of sets, there exists a categorical product in the category $\mathbf{Set}$ of sets, does this imply that the set-theoretic construction $\left\{(x_i)_{i\in I}\in\left(\bigcup_{i\in I}X_i\right)^I\middle|x_i\in X_i\text{ for all $i$}\right\}$ is non-empty, i.e. that the Axiom of Choice holds?

Answer 1

No -- in a world without choice it may simply be that $\varnothing$ satisfies the conditions for being the product object.

It would do so almost vacuously: In order to apply the universal property of products you would need to have an object $Y$ and a family of morphisms $Y\to X_i$; but if $(X_i)_i$ doesn't have a choice function, then such a family of morphisms can't exist unless $Y$ happens to be the empty set.