Say we have category $C$ with class of morphisms weak equivalence and cofibration. And say we have that (Acylic) cofibrations are closed under pushouts, composition. And that initial object is in $C$ and every object is cofibrant and isomorphism are weak equivalences and weak equivalences satisfy 2-out-of-6.
With this, to check that every map can be factored as cofibration followed by weak equivalence, it is a fact that it suffices to check the factorization for the codiagonal map $X \sqcup X \to X$.
I'm looking for the proof of this fact.
Thanks!