composition of pointed homotopies

Let \((X,x_0), (Y,y_0), (Z,z_0)\) be based spaces and

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

be pointed homotopic based maps. Then

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

are pointed homotopic

corollary of

• precomposition of homotopic maps
• postcomposition of homotopic maps

where \(x_0 \in g^{-1}[y_0]\)