The more general question is whether pullbacks and colimits commute. This is true of if the category $\mathcal C$ is locally cartesian closed, i.e. every slice category $\mathcal C /c$ of objects over a given object $c$ of $\mathcal C$ is cartesian closed. For then pullback has a right adjoint.
These ideas are of interest in the topological case, though even if $\mathcal C$ is a convenient category of spaces, the categories $\mathcal C/c$ are not always cartesian closed.
Nonetheless, the use of such categories was initiated by R. Thom and continued by Peter Booth, see for example:
Booth, Peter I. A unified treatment of some basic problems in homotopy theory. Bull. Amer. Math. Soc. 79 (1973), 331–336.
and his related papers: these ideas were also taken up by Ioan James under the term "Fibrewise topology", see
Crabb, Michael; James, Ioan; Fibrewise homotopy theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 1998.
There is a paper
Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.
giving a locally cartesian closed category, in fact a topos, with sequential spaces as a reflective subcategory, but this has not yet been used in algebraic opology, to my knowledge.