Proposition: Exact reconstruction descent

Let $P,K,Y$ be vector spaces, $\mathrm{Rec}:P\to K$ a surjective linear map, and $\tilde{M}:P\to Y$ linear. A unique linear map $M:K\to Y$ with $\tilde{M}=M\mathrm{Rec}$ exists if and only if

If $K$ has the quotient topology induced by $\mathrm{Rec}$ and $\tilde{M}$ is continuous, the descended map is continuous.

Proof. Necessity follows by applying $M\mathrm{Rec}$ to a null presentation. For sufficiency, define $M(\mathrm{Rec}p)=\tilde{M}(p)$. Presentations with equal reconstruction differ by an element of $\ker \mathrm{Rec}$, so the definition is independent of the presentation. Surjectivity gives uniqueness. The quotient topology gives continuity.