homotopy under composition

Let \((X, \mathcal{T}_X)\), \((Y, \mathcal{T}_Y)\) and \((Z, \mathcal{T}_Z)\) be topological spaces, \(f_0,f_1: X \rightarrow Y\) and \(g_0,g_1: Y \rightarrow Z\) continuous map such that

\begin{align*} f_0 \cong& f_1 \\ g_0 \cong& g_1 \end{align*}

Then

\begin{align*} g_0 \circ f_0 \cong g_1 \cong f_1 \end{align*}

Special case of relative homotopy under composition (see: relative Homotopie bezüglich der leeren Menge)