Kerodon

In definition 5.3.3.1 I believe $\operatorname{N}^{\mathscr{F}}n(C)$ be $\operatorname{N}^{\mathscr{F}}(C)_n$.

Sorry, my previous comment was incorrect. I should have said $\operatorname{N}^{\mathscr{F}}_n(C)$ (so that there's just a missing _).

Yep. Thanks!

In the first paragraph the same map is called $U$ and later $V$.

Yep. Thanks!

?62? $\mathcal{F}$ carries each morphism of $\mathcal{C}$ to a Kan fibration, then the projection map $U$ is a left fibration of simplicial sets.

I think it also needs to assume that $\mathcal{F}(C)$ is a Kan complex for each $C\in \mathcal{C}$?

If $\mathscr{F}$ is a functor from $\calC$ to $\mathrm{Kan}$, then (by definition) it carries each object of $\calC$ to a Kan complex.