Proposition: Descent through reconstruction

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

If $\mathcal{K}$ carries the quotient topology and $\tilde{\mathsf{M}}$ is continuous, the induced map is continuous.

Proof. Factorization immediately implies (eq:descent). Conversely, define $\mathsf{M}(\mathsf{Rec}A)=\tilde{\mathsf{M}}(A)$. Two presentations of the same kernel differ by an element of $\ker \mathsf{Rec}$, so the definition is independent of presentation. Surjectivity gives uniqueness. Continuity is the defining property of the quotient topology.