# Right Lifting Properties Against Relative Cell Complexes

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.

• Every morphism in $K \textrm{-Inj}$ has the rlp with respect to $K$. Consider $(K \textrm{-Inj})\textrm{-Proj}$. Every morphism in $K \textrm{-Inj}$ has the rlp with respect to $(K \textrm{-Inj})\textrm{-Proj}$, and $K \subseteq (K \textrm{-Inj})\textrm{-Proj}$. Apply the proposition. – Zhen Lin Aug 17 '20 at 22:34
• Thanks! If want you should answer this question so I can accept it! – Pedro Brunialti Aug 17 '20 at 23:18

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.