Proposition: Descent through a reconstruction map

Let $\mathsf{Rec}:P\to K$ be a surjective linear reconstruction map. A linear candidate $F:P\to V$ has the form $F=M\mathsf{Rec}$ for a map $M:K\to V$ if and only if

When it exists, $M$ is unique.

Proof. If $F=M\mathsf{Rec}$, every reconstruction-null element has zero image. Conversely, equal reconstructions have a difference in $\ker \mathsf{Rec}$. The kernel condition therefore makes $M(\mathsf{Rec}p)=F(p)$ well defined. Surjectivity gives uniqueness.