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The following is not an answer, just an observation suggesting that maybe the question should be rephrased. I claim that if $X$ is a fibrant scaled simplicial set whose decalage is a fibrant marked si …

Yes. Since $X^{\natural} \to S$ and $Y^{\natural} \to S$ are both cartesian fibrations they are fibrant and cofibrant objects in the cartesian model structure over $S$, which is a simplicial model str …

Yes, though it is usually written as the commutativity of unstraightening with pullback (on the $\infty$-categorical level it doesn't matter, since straightening and unstraightening are inverse equiva …

No. If this were the case then there would be a section $s: B \to E$ to $f$ induced by the $G$-equivariant map $\widetilde{s}:\widetilde{B} \to \widetilde{B} \times {\rm K}(M,n)$ sending $x$ to $(x,0) …

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from t …

I think what Lurie might have meant when he wrote "It is easy to see that $St_{\phi}$ preserves cofibrations" in the proof of Theorem 2.2.1.2, is that it is easy to see it if you take into account the …

Concerning the first question: the simplicial localization functor $L^H$ induces an equivalence from the relative category of small relative categories to the relative category of small simplicial cat …