Proposition: Descent criterion

A linear map $\tilde{\mathsf{M}}:{\mathcal{P}}_{\mathrm{int}}\to {\mathcal{D}}^{\prime }(U)$ defines a unique map on ${\mathcal{K}}_{\mathrm{int}}$ exactly when

Continuity of the lifted map gives continuity of the descended map for the quotient topology.

Proof. Two presentations have the same reconstruction exactly when their difference belongs to $\ker \mathsf{Rec}$. Annihilating this kernel makes the proposed image independent of the representative. The quotient universal property proves the continuity statement.