Proposition: Presentation descent

A linear map $\tilde{M}:\mathcal{P}\to {\mathcal{D}}^{\prime }(W)$ defines a unique map $M:\mathcal{K}\to {\mathcal{D}}^{\prime }(W)$ with $\tilde{M}=M\mathrm{Rec}$ if and only if

Proof. Factorization annihilates the reconstruction kernel. Conversely, two equal reconstructions differ by an element of that kernel, so the condition makes their images equal and defines the unique induced map.