Proposition: Actual-functional descent

Let $\mathsf{Rec}:\mathcal{P}\to \mathcal{K}$ be a surjective linear presentation map. A proposed linear construction $\tilde{M}:\mathcal{P}\to Y$ descends to $\mathcal{K}$ if and only if

With the quotient topology, a continuous such map descends continuously.

Proof. Necessity follows by testing a null presentation. Conversely, two presentations of the same element differ by a reconstruction-null element, so their images agree. Surjectivity gives uniqueness, and the quotient topology gives continuity.