Lemma: Factorization through declared data

For maps $\pi :D\to B$ and $F:D\to V$, a map $\overline{F}:\pi (D)\to V$ satisfying $F=\overline{F}\circ \pi$ exists if and only if $F$ is constant on each fiber of $\pi$.

Proof. A composition with $\pi$ is constant on fibers. Conversely, define $\overline{F}(\pi (d))=F(d)$. Fiber constancy makes this definition independent of $d$.