Lemma: Reconstruction compatibility

For a surjective linear reconstruction $\mathsf{Rec}:P\to K$, a linear $F:P\to V$ descends to $K$ if and only if

The induced map is unique.

Proof. Necessity follows from $F=M\mathsf{Rec}$. For sufficiency, define $M(\mathsf{Rec}p)=F(p)$. Equal reconstructions have a difference in $\ker \mathsf{Rec}$, so the definition is independent of the representative. Surjectivity proves uniqueness.