Proposition: Descent through reconstruction

Let

be a surjective linear reconstruction map, and let $\tilde{M}:\mathcal{P}\to \mathcal{Y}$ be linear. There is a unique map $M:\mathcal{K}\to \mathcal{Y}$ with $\tilde{M}=M\mathsf{Rec}$ if and only if

Proof. Factorization annihilates every reconstruction-null presentation. Conversely, define $M(\mathsf{Rec}A)=\tilde{M}(A)$. Two presentations of the same reconstruction differ by an element of $\ker \mathsf{Rec}$, so the definition is independent of the presentation. Surjectivity proves uniqueness.