Holmstrom Left homotopy

Let $f,g: B \to X$ be morphisms in a model category. A left homotopy from $f$ to $g$ is a map $H: B' \to X$ for some cylinder object $B'$ for $B$ such that $H \circ i_0 = f$ and $H \circ i_1 = g$. Here $i_0$ and $i_1$ are the obvious maps. Say $f$ and $g$ are left homotopic if there exists such a thing.

nLab page on Left homotopy