Question about Lemma 7.7.1 from Hirschhorn's Model Categories and Their Localizations The Lemma states the following.
Let M be a model category. If $g:X\rightarrow Y$ is a weak equivalence between cofibrant objects in M, then there is a functorial factorization of $g$ as $g=ji$ where $i$ is a trivial cofibrations and $j$ is a trivial fibration that has a right inverse that is a trivial cofibration.
Can one clarify what functorial means here and how to show it? For me, it seems that this part is not proven in the proof of Lemma?
 A: The meaning of "functorial factorization" is well clarified in the following article
http://arxiv.org/pdf/1204.5427.pdf
at page 7.
What you require is the existence of two factorization functors $R, L : \mathrm{Mor}(M) \to \mathrm{Mor}(M)$ which can be used to factor any morphism, and are well behaved with respect to domains and codomains. That is: $\mathrm{dom} (L) = \mathrm{dom}, \mathrm{cod}(L)=\mathrm{dom}(R), \mathrm{cod}(R) = \mathrm{cod}$ as functors. For example, $\mathrm{dom}$ is a functor $\mathrm{Mor}(M) \to M$, defined in the only way that makes sense; $\mathrm{dom}(R)$ denotes the composite functor $\mathrm{dom} \circ R$, with a little abuse of notation. Here $\mathrm{Mor}(M)$ is the category of morphisms in $M$ and commutative squares between them. In the article I linked it is called $M^2$.
Functorial factorizations are given in any model category, as an axiom. Some authors, such as Hovey in his book "Model Categories", include the functorial factorization in the model structure itself, not merely requiring their existence.
