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Is it true that when the empty family of arrows to an object $E$ in some category is epimorphic, that object $E$ must be the initial object $0$?

This is a claim on page 433 (eq. 22) of Mac Lane and Moerdijk - Sheaves in Geometry and Logic, in the proof of Theorem VIII.2.7.

I can see that the assumptions imply that any map from $E$ to any other object $X$ is necessarily unique. Indeed, assuming that the ambient category has coproducts, the empty family being epimorphic means that the map $e$ from the empty coproduct (i.e. the initial object $0$) is an epimorphism. Now given two maps $f,g: E\to X$, we have $fe=ge$ is the unique map $0\to X$, hence $f=g$ since $e$ is epic.

However I cannot see why there should exist a map $E\to X$ which would make $E$ initial.

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    $\begingroup$ (Actually, no need for coproducts: the empty family to $E$ being epimorphic just means that by definition.) Existence is not guaranteed. Cook up a simple counterexample with 2 or 3 objects.. Relevant: math.stackexchange.com/questions/1092122/…. $\endgroup$ – Berci May 29 '16 at 23:48
  • $\begingroup$ Note that the referred claim in MacLane and Moerdijk is not needed for the proof to carry through. All the proof requires is that if $E=0$ then the empty family is epimorphic. The claim is the converse statement, which does not impact the soundness of the proof. $\endgroup$ – Adrien Vakili May 31 '16 at 0:13
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This is not true in general. Indeed, your argument can be reversed to show that whenever $E$ is an object with at most one map to every object, then the empty family is epimorphic onto $E$. But there are many categories with objects that have at most one map to every object but do not map to every object (for instance, any nontrivial poset).

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