# Perfect class of morphisms closed under retracts?

Suppose we are working in a presentable class of morphisms $$\mathcal{C}$$. A class of morphisms $$P$$ is said to be perfect if :

1) The class $$P$$ contains all isomorphisms.

2) The class $$P$$ statisfies the two-out-of-three property.

3) It is close under filtered colimits.

4)There exists a set $$P_0 \subseteq P$$ such that every morphism in $$P$$ can be obtained as a filtered colimit of morphisms in $$P_0$$.

The question is then: is perfect class of morphisms closed under retracts? I've stumbled upon some occurences in which it looks like they are using this fact but I can not manage to produce a proof. I was thinking that using the two first axioms it would be simple to show it but I am stuck...

It doesn't follow from the first two axioms. Instead, you must use that any retract can be written as a filtered colimit. Given a split epimorphism $$(h,k): f\to g$$ in the arrow category with splitting $$(i,j)$$, we may write $$g$$ as the colimit of the filtered diagram $$f\to f\to f\to...$$, where every map is $$(ih,jk)$$. This isn't special to the arrow category, and uses none of properties 1,2,4: any category closed under filtered colimits is closed under retractions.