Proposition: Exact descent criterion

Let $\mathsf{Rec}:P\to K$ be a surjective linear reconstruction map. A linear map $\tilde{M}:P\to Y$ factors as $\tilde{M}=M\mathsf{Rec}$ if and only if

When it factors, $M$ is unique.

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