Lemma: Reconstruction descent

Let $\mathsf{Rec}:P\to K$ be a surjective linear map. A linear candidate $\tilde{M}:P\to D$ factors uniquely through $K$ if and only if

Proof. Necessity follows from factorization. Conversely, define $M(\mathsf{Rec}p)=\tilde{M}(p)$. Presentations of the same reconstruction differ by a null presentation, so this is well defined and unique.