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.