Final-discrete-fibration factorization
= Final-discrete-fibration factorization
Every functor factors as a <final functor> followed by a <discrete fibration>. For $F:\mathcal C\to\mathcal D$, the intermediate category has objects $(B,c)$, where $B\in\mathcal D$ and $c$ is a connected component of $(B\downarrow F)$. An arrow $(B,c)\to(B',c')$ is an arrow $g:B\to B'$ whose precomposition functor sends $c'$ to $c$.