Homotopy pushout of spaces

Definition 2.1.3. Consider maps \(f\colon Z \to X\) and \(g\colon Z \to Y\) of topological spaces. Their homotopy pushout \(X \sqcup _Z^h Y\) (or double mapping cylinder) is defined as \begin {align*} X \sqcup _Z^h Y \quad &:= \quad X \sqcup _Z (Z \times [0,1]) \sqcup _Z Y, \end {align*}

where the right-hand side is the quotient of \(X \sqcup (Z \times [0,1]) \sqcup Y\) by the equivalence relation generated by \((z,0) \sim f(z)\) and \((z,1) \sim g(z)\) for all \(z \in Z\). We denote by \(\iota _X\colon X \hookrightarrow X \sqcup _Z^h Y\) and \(\iota _Y\colon Y \hookrightarrow X \sqcup _Z^h Y\) the canonical inclusions. There is also a canonical homotopy \[ H_{\can }\colon \iota _X \circ f \sim \iota _Y \circ g, \] represented by the map \(Z \times [0,1] \to X \sqcup _Z^h Y\) sending \((z,t)\) to \([(z,t)]\).