I have trouble understanding the usage of homotopy pushouts in the proof of proposition 2.1.4.10 in Lurie’s Higher Topos Theory. My trouble is the way he shows that for a map $jcolon Sto S’$, the induced functor $j_!colon (mathcal{Set}_Delta)_{/S}to(mathcal{Set}_Delta)_{/S’}$ preserves weak equivalences in the covariant model structure.

Given a covariant equivalence $fcolon Xto Y$ (over $S$), I want to show that $j_!f$ is a covariant equivalence as well. This seems to be the relevant diagram. The top horizontal arrow is induced by $f$ and hence by assumption an equivalence of simplicial categories. The bottom horizontal arrow is $j_!f$. From what I understand, the diagram Lurie draws (one of the backsides of my diagram) is a pushout. This follows from the pasting laws for pushouts and the facts that $mathfrak{C}(-)$ preserves colimits. He then states that $mathfrak{C}(-)$ preserves monomorphisms and hence (I believe by A.2.4.4) he deduces that it must be a homotopy pushout as well. According to Lurie we are done.

Here are my questions:

- Why does this imply that the map $j_!f$ (the bottom horizontal map in my diagram) is an equivalence?
- Is my assertion correct that he uses A.2.4.4 to deduce that the square is a homotopy pushout? I don’t really see the relevant cofibration.
- In a similar vein he mentions in the proof of 2.1.4.7 (2) that a square is a homotopy pushout to conclude that the bottom arrow in a sqaure is an equivalence. This is equally confusing to me: Do homotopy pushouts preserve weak equivalences in general?

Any hint would be helpful, so thanks in advance.