Kerodon

Warning 5.5.3.9. The coslice comparison functor $F: \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } ) \rightarrow \operatorname{\mathcal{S}}_{\ast }$ of Proposition 5.5.3.8 is bijective on vertices: objects of either $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Kan}_{\ast } )$ and $\operatorname{\mathcal{S}}_{\ast }$ can be identified with pointed Kan complexes $(X,x)$. However, it is not bijective on edges (and is therefore not an isomorphism of simplicial sets). If $(X,x)$ and $(Y,y)$ are pointed Kan complexes, then a morphism from $(X,x)$ to $(Y,y)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$ can be identified with a pair $(f,h)$, where $f: X \rightarrow Y$ is a morphism of Kan complexes and $h: f(x) \rightarrow y$ is an edge of the Kan complex $Y$. The pair $(f,h)$ belongs to the image of $F$ if and only if the edge $h$ is degenerate (which guarantees in particular that $f(x) = y$, so that $f$ is a morphism of pointed Kan complexes).