Kerodon

It seems that there are some typos in the definition of $S$ in the proof. I think there's a a $\operatorname{\mathcal{D}}$ that should be $\operatorname{\mathcal{C}}$, and the $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ of the last fiber product are swapped. So it should say: "Let S be the collection of all monomorphisms of simplicial sets $f: A \hookrightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$, the induced map $A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a categorical equivalence."

Sorry, in my last comment I meant the proof of Lemma 5.3.6.5 (posted it in the wrong place).

Yep, thanks!