Remark 4.5.6.25 (05J0)

Remark 4.5.6.25. For every simplicial set $S$, let $\underline{S}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ denote the functor which carries each object $[m] \in \operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ m, S)$, regarded as a constant simplicial set. Unwinding the definitions, we see that the morphism $\theta _ n$ appearing in Proposition 4.5.6.23 can be identified with the restriction map

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \underline{ \Delta ^ n}, \mathscr {F} )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{{\bf \Delta }}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \underline{ \operatorname{\partial \Delta }^ n}, \mathscr {F} )_{\bullet }. \]

Consequently, if $\mathscr {F}$ is isofibrant, then $\theta _ n$ is an isofibration of $\infty $-categories (Proposition 4.5.6.11 and Corollary 4.5.6.12). If $\mathscr {F}$ is an isofibrant diagram of Kan complexes, then $\theta _ n$ is a Kan fibration between Kan complexes (Corollary 4.5.6.20).