Homotopy invariance of homotopy pushouts

Exercise 2.1.4. Show that the homotopy type of \(X \sqcup _Z^h Y\) depends only on the homotopy types of \(X, Y, Z\) and the homotopy classes of \(f, g\). In more detail, consider a homotopy commutative diagram of topological spaces and continuous maps of the form

where \(H\colon \alpha \circ f \sim f' \circ \gamma \) and \(K\colon \beta \circ g \sim g' \circ \gamma \) are homotopies. Construct a map \(\alpha \sqcup _{\gamma }^h \beta \colon X \sqcup _Z^h Y \to X' \sqcup _{Z'}^h Y'\), and show it is a homotopy equivalence whenever \(\alpha \), \(\beta \) and \(\gamma \) are homotopy equivalences.