Connected-component presheaf of a functor ($B\mapsto {\pi }_0(B\downarrow F)$)

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.