Proposition: Continuous reconstruction descent

A continuous linear $G:P\to {\mathcal{D}}^{\prime }(X;E)$ factors uniquely through a continuous map on $K$ if and only if $G(\ker \mathsf{Rec})=0$.

Proof. Define the descended map by $M(\mathsf{Rec}A)=G(A)$. Null annihilation makes it well defined; the quotient topology gives continuity.