In the first chapter of his book "Model Categories" Mark Hovey repeatedly makes use of the following without proof.
Our setting is that of a model category. Let $B\sqcup B \overset{i_0+i_1}{\longrightarrow} B' \overset{s}{\longrightarrow} B$ be a cylinder object for $B$, i.e. $i_0+i_1$ is a cofibration, $s$ is a weak equivalence and $s \circ (i_0+i_1) = \bigtriangledown$, where $\bigtriangledown$ is the fold map. If $B$ is cofibrant then $B \overset{i_1}{\longrightarrow} B'$ is a trivial cofibration.
Here $i_0 + i_1$ is the map induced by $i_0: B\to B'$ and $i_1:B \to B'$ and $\bigtriangledown = 1_B + 1_B$.
I can see why in any case (even without assuming $B$ cofibrant) $i_1$ has to be a weak equivalence (by the 2 out of 3 property), but how does it follow that it is a cofibration?
Best Answer
Both inclusion arrows from $B$ to the coproduct are pushouts of $0\to B$. So if a sum is a cofibration and $B$ is cofibrant, both components are cofibrations.