Pushout commutes with $- \times I$

Question

Consider top spaces $A, X, Y$ und morphisms $f:A \to X, g:A \to Y$.

How can I see (prefered by abstract/ category theoretical argument) that we have isomorphism $$(X \cup_{f,g} Y) \times I \cong (X \times I) \cup_{f \times id ,g \times id} (Y \times I) $$

where $B \cup_{f,g} C$ is the induced pushout and $I = [0,1]$ the canonical intervall.

Answer 1

That functor is a left adjoint, so it commutes with colimits.