Proposition 2

Let X, Y and Z be topological spaces, $f$ and $f'$ continuous mappings of X into Y$,g$ and $g'$ continuous mappings of Y into Z. If $f$ is homotopic to $f'$ and if $g$ is homotopic to $g'$, then $g\circ f$ is homotopic to $g'\circ f'$.

Let $\sigma$ be a homotopy joining $f$ to $f'$ and $\tau$ a homotopy joining $g$ to $g'$. Then the mapping $\theta : X\times \mathbf{I}\rightarrow Z$ defined by $\theta (x, t) =\tau (\sigma (x, t), t)$ is a homotopy joining $g\circ f$ to $g'\circ f'$.

Let X, Y and Z be topological spaces. Given homotopy classes $\varphi \in [X; Y]$ and $\psi \in [Y; Z]$, the mappings $g\circ f: X\rightarrow Z$, where $f\in \varphi ,g\in \psi$, all belong to one and the same homotopy class (Prop. 2), which is denoted by $\psi \circ \varphi$ and is called the composite homotopy class of the classes $\psi$ and $\varphi$. The mapping $[X; Y]\times [Y; Z]\rightarrow [X; Z]$ which associates $\psi \circ \varphi$ to $(\varphi , \psi )$ is called the composition mapping.

Let $\varphi \in [X; Y]$ ; we have $\varphi =\varphi \circ$ [Id$_X] =$ [Id$_Y]\circ \varphi$.

Let X, Y, Z and T be topological spaces, let $\varphi \in [X; Y]$, $\psi \in [Y; Z]$ and $\chi \in [Z; T]$. The homotopy class $\chi \circ (\psi \circ \varphi )$ is equal to $(\chi \circ \psi )\circ \varphi$; it is denoted by $\chi \circ \psi \circ \varphi$.