Right Lifting Properties Against Relative Cell Complexes

Question

So I've been studying these notes on homotopy theory. There is a proposition (2.10) which states that for any collection of morphisms $K \subset \mathrm{Mor}(C)$ the collection KProj of $K$-projective morphisms and KInj of $K$-injective morphisms satisfy the following

$\bullet$ Both classes are closed under composition and KProj is closed under transfinite composition.

$\bullet$ Both classes are closed under forming retracts in the arrow category of $C$.

$\bullet$ KProj is closed under forming pushouts of morphisms in $C$ and KInj is closed under forming pullback of morphisms in $C$.

$\bullet$ KProj is closed under forming coproducts in in the arrow category of $C$ and KInj is closed under forming products in the arrow category of $C$.

As a corollary of such proposition we have:

Let $C$ be a category with all small colimits, and let K⊂Mor($C$) be a sub-class of its morphisms. Then every K-injective morphism has the right lifting property against all K-relative cell complexes and their retracts.

The problem is that I can't see why this corollary follows, I've tried using the universal properties of the pushout in the transfinite composition of a cell complex but it doesn't seem to work, maybe I'm just not seeing something? Any help is appreciated.

Answer 1

Every morphism in $K\textrm{-Inj}$ has the right lifting property with respect to $K$. Consider $(K\textrm{-Inj})\textrm{-Proj}$. Every morphism in $K\textrm{-Inj}$ has the right lifting property with respect to $(K\textrm{-Inj})\textrm{-Proj}$, and $K \subseteq (K\textrm{-Inj})\textrm{-Proj}$. The proposition says $(K\textrm{-Inj})\textrm{-Proj}$ is closed under pushouts, retracts, and transfinite composition. The claim follows.