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 $c$.

Two factorizations of one functor have a unique connecting functor in each direction, by the orthogonality of final functors and discrete fibrations. Their composites and the identity fill the same lifting square, so uniqueness makes both composites identities. The intermediate categories are therefore canonically isomorphic.

For a functor $F:\mathcal{C}\to \mathcal{D}$, precomposition by $u:B\to B^{\prime }$ induces $(B^{\prime }\downarrow F)\to (B\downarrow F)$, hence a map of categorical connected components in the opposite direction to $u$. This defines a presheaf on $\mathcal{D}$. Its category of elements is the intermediate category in the final-discrete-fibration factorization.