In any category where the objects are sets equipped with certain relations and operations, the notion of "morphism" arises perfectly naturally. (Generally, a morphism between objects is one that commutes with the operations, and maps tuples in a relation to tuples in a relation.)
On the other hand, the definition of a morphism from topological space $(X, \tau_X)$ to $(Y, \tau_Y)$ is of course a continuous map $f$, i.e. a map such that the inverse image of $f$, $$ f^{-1} : \mathcal{P}(Y) \to \mathcal{P}(X) $$ preserves open sets, rather than $f$ itself preserving open sets.
Therefore my (very basic!) question is: in what sense is taking the morphisms to be the continuous maps natural or canonical? I'm assuming that there is some a priori reason that we might expect the continuous maps to be a better definition of morphism than, say, functions such that the image of an open set isn't an open set.
Perhaps an answer comes from pointless topology, or from a better understanding of the correspondence between a function $f: X \to Y$ and its inverse image $f^{-1}: \mathcal{P}(Y) \to \mathcal{P}(X)$.