In Lemma 8.5 (chapter II) of Goerss & Jardine's book, we are given the following pushout square in a category of cofibrant objects:
$$\begin{array}{cc} A & \xrightarrow {u} & B \\ \downarrow{i} & & \downarrow \\ C & \xrightarrow{u_*} & D \end{array}$$
where $i$ is a cofibration and $u$ is a weak equivalence. The lemma claims that $u_*$ is a weak equivalence.
I did not understand the first and the last parts of the argument. At the start of the proof, they claim
Trivial cofibrations are stable under pushouts, so Lemma 8.4 implies that it suffices to assume that there is a trivial cofibration $j:B\to A$ such that $u\cdot j=1_B$.
Lemma 8.4 claims that any map in a category of cofibrant objects can be factorised into a cofibration and a weak equivalence that is left inverse to a trivial cofibration.
I am unable to see how lemma $8.4$ can be used to assume that $u\cdot j= 1_B$.
I don't understand the last paragraph of the proof as well (where the authors switch to a slice category). Any help will be appreciated.
The cited lemma implies $u : A \to B$ can be factored as a trivial cofibration $a : A \to A'$ followed by a weak equivalence $u' : A \to B$ that is left inverse to a trivial cofibration $j' : B' \to A$. The pushout of a composite is the composite of the pushouts, and we already know that the pushout of a (trivial) cofibration is a (trivial) cofibration, so we may reduce to the case where $u : A \to B$ itself is left inverse to a trivial cofibration.