Discrete fibration

A functor $F:\mathcal{C}\to \mathcal{D}$ is a discrete fibration when every arrow $B\to FA$ has a unique lift with codomain $A$.

An incoming morphism into the image of a specified object has exactly one lift with that specified codomain. Both its source object and its lifted morphism are determined. Applying uniqueness to identity and composite morphisms gives the functoriality of lifting.

Given a commutative square whose left functor is final and whose right functor is a discrete fibration, there is a unique diagonal functor filling the square. To construct its value on an object, choose an object of the relevant connected comma category and lift the resulting arrow; uniqueness of lifts makes the answer constant over that connected category.

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.