Anyone know if anyone has proven that the Koszul duality stuff of Ayala-Francis is compatible with the twisted arrow category stuff of Lurie?

Hi, I'm new. I was wondering if someone has a counterexample or a proof of the following statement: Let X and Y be two simplicial groups such that there exists f:X>----->Y and g:Y >----> X two injective morphisms of simplicial groups. Suppose that X is contractible, does it follow that Y is contractible ? Thank you in advance

Probably idiot. Suppose an adjunction $F \dashv G : C \rightarrow C$. Now, let $Sp (C) = \lim (... \rightarrow C \rightarrow C)$ (the arrows are $G$ here). Without assuming that $C$ is a locally presentable quasicategory, is it true that $Sp (C) \cong colim (C \rightarrow C ...)$? It seems that it's true, but I see in Robalo's thesis only the remark for the locally presentable case.