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 $(B,c)\to (B^{\prime },c^{\prime })$ is an arrow $g:B\to B^{\prime }$ whose precomposition functor sends $c^{\prime }$ to $c$.