Lemma: Presentation descent

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

Proof. Necessity follows by applying the factorization to a null presentation. Conversely, define $M(\mathsf{Rec}p)=\tilde{M}(p)$. Two presentations of the same reconstructed object differ by an element of $\ker \mathsf{Rec}$, so the definition is well defined. Surjectivity gives uniqueness.