homotopy inverse

Let \((X,\mathcal{T}_X)\) and \((Y,\mathcal{T}_Y)\) be topological spaces and \(f: X \rightarrow Y\) a Homotopy equivalence. Then a continuous map \(g: Y \rightarrow X\) is said to be a homotopy inverse, if

\begin{align*} \mathrm{id}_{X} \sim& g \circ f \\ \mathrm{id}_{Y} \sim& f \circ g \end{align*}