Universal property of the pushout and pasting lemma If $A,B$ are closed subspaces of a topological space $X$ with $X=A\cup B$. How can we express the pasting Lemma for these closed sets as a universal pushout property?

Given topological spaces $Y$ and $Z$ let $\hom(Y, Z)$ be the set of continuous maps between them. Then the pasting lemma says that for all topological spaces $Z$ the diagram $$\begin{matrix} \hom(X, Z) & \longleftarrow & \hom(B, Z) \\\ \uparrow && \uparrow \\\ \hom(A, Z) & \longleftarrow & \hom(A\cap B, Z)\end{matrix}$$ with maps induced by the inclusions, is a pushout in the category of sets.