Proper maps between pairs

Question

If $(X,A)$ and $(Y,B)$ are pairs of topological spaces, what would be a good notion of a proper map $f: (X,A) \to (Y,B)$?

Maybe $f:X\to Y$ is proper and $f|_A: A\to B$ is proper?

Answer 1

In topology, a proper map f:X -> Y is a map with
for all U compact subset Y, $f^{-1}(U)$ is compact.

As A is not a subset of X, it makes no sense to
restrict f to A. What notion of proper are you
using? Something like proper subset?