composition of homotopic maps

Let \((X, \mathcal{T}_X)\), \((Y, \mathcal{T}_Y)\) and \((Z, \mathcal{T}_Z)\) be topological spaces and

\begin{align*} f,f': X \rightarrow& Y \\ g,g': Y \rightarrow& Z \end{align*}

homotopic relative to the empty set. Then for a map \(h: X \rightarrow Y\), it follows, that

\begin{align*} g \circ h \circ f \sim g' \circ h \circ f' \end{align*}

are homotopic relative to \(\emptyset\)

corollary of:

• postcomposition of homotopic maps
• precomposition of homotopic maps

for \(\mathrm{id}_{Y}\)