The Clowder Project

Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.

1. Characterisations. The following conditions are equivalent:^[1]
   1. The functor $F$ is a epimorphism of categories.
   2. For each morphism $f\colon A\to B$ of $\mathcal{D}$, we have a diagram
      
      in $\mathcal{D}$ satisfying the following conditions:
      1. We have $f=\alpha _{0}\circ \phi _{1}$.
      2. We have $f=\psi _{m}\circ \alpha _{2m}$.
      3. For each $0\leq i\leq 2m$, we have $\alpha _{i}\in \textup{Mor}\webleft (\mathrm{Im}\webleft (F\webright )\webright )$.
2. Surjectivity on Objects. If $F$ is an epimorphism of categories, then $F$ is surjective on objects.