Corollaire

Let $C, C', C''$ be three complexes, $f$ and $g$ two morphisms of $C$ into $C'$, $f_1$ and $g_1$ two morphisms of $C'$ into $C''$. If $s$ and $s_1$ are homotopies relating $f$ to $g$ and $f_1$ to $g_1$ respectively, then $s_1 \circ f + g_1 \circ s$ is a homotopy relating $f_1 \circ f$ to $g_1 \circ g$. If $f$ and $f_1$ are homotopic to $g$ and $g_1$ respectively, then $f_1 \circ f$ is homotopic to $g_1 \circ g$. Indeed, $s_1 \circ f$ relates $f_1 \circ f$ to $g_1 \circ f$ and $g_1 \circ s$ relates $g_1 \circ f$ to $g_1 \circ g$.