I have some conditional questions. First, in the category of topological pairs (as used in cohomology theories), are these only pairs of the form $(X,A)$ with $A \subset X$?
If the answer to this question is yes, then my next question is: is a morphism $(X,A) \to (Y,B)$ a map $f:X \to Y$ with $f(A) \subset B$ or pairs of maps $f_1: X \to Y$ and $f_2: A \to B$?
Thanks!