A set-function between spaces is proper if pulls back compacts to compacts, closed if it pushes closed sets to closed sets, and universally closed if all its pullbacks are closed.
I am wondering if and when continuous functions glue along (continuous) surjective proper maps and/or (continuous) surjective universally closed maps. Specifically, I am wondering when the diagram $$\mathrm h_Z(Y)\to \mathrm h_Z(X)\rightrightarrows \mathrm h_Z (X\times _Y X)$$ induced from the kernel pair diagram of $f:X\to Y$ is an equalizer diagram given that $f$ is (continuous and) proper and/or (continuous and) universally closed.
(Here $\mathrm h_Z=\mathrm{Hom}(-,Z):\mathsf{Top}^\text{op}\to \mathsf{Set}$.)