Considering each half of factorization of weak equivalence separately

I have been working through the proof of Theorem 3.4.1 in the article https://arxiv.org/pdf/1211.2851.pdf (pages 35-6), but there is one technical detail still unclear to me.

Specifically, we have constructed a commutative diagram in $$\mathbf{sSet}$$ such that

1. $$A \to B$$ is a cofibration,
2. $$w$$ is a weak equivalence,
3. $$E_i\to A$$ is a Kan fibration, and
4. $$\overline{E}_2 \to B$$ is a Kan fibration.

We now want to prove the claim (c) that $$\overline{E}_1$$ is a fibration over $$B$$ and $$\overline{w}$$ is a weak equivalence. The authors state that by factoring the weak equivalence $$w$$ as a trivial cofibration followed by a trivial fibration, we may prove (c) assuming that $$w$$ is either a trivial cofibration or a trivial fibration. I'm sure this is obvious to most, but I can't see why this suffices. I would be grateful for an explanation.

Note that the construction $$w \mapsto \overline{w}$$ is functorial. Thus, if $$w = v u$$ is a factorization as a trivial cofibration followed by a trivial fibration, we have $$\overline{w} = \overline{v} \,\overline{u}$$.
Now if (c) holds under the two stated assumptions, the statement of (c) can be applied to $$u$$ and $$v$$. Applied to $$v$$, we see that $$\overline{v}$$ is a weak equivalence and its domain is a fibration over $$B$$. The latter conclusion now means that we can apply (c) to $$u$$, concluding that $$\overline{u}$$ is a weak equivalence and its domain is a fibration over $$B$$. But the domain of $$\overline{u}$$ is $$\overline{E_1}$$, and $$\overline{w} = \overline{v} \,\overline{u}$$ so it is also a weak equivalence.