Kerodon

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Notation 8.2.1.2. Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.6.1.1). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and suppose we are given a pair of functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. Fix an object $X \in \operatorname{\mathcal{C}}$ and a vertex $\eta \in \mathscr {F}(X)$. We then obtain a comparison morphism

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \mathscr {F}, \mathscr {G} ) & \xrightarrow { \operatorname{ev}_{X} } & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \mathscr {F}(X), \mathscr {G}(X) ) \\ & \xrightarrow { \circ [\eta ] } & \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, \mathscr {G}(X) ) \\ & \simeq & \mathscr {G}(X) \end{eqnarray*}

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the first map is given by evaluation on the object $X$, the second by the composition law of Notation 4.6.8.15, and the third is (the inverse of) the homotopy equivalence of Remark 5.6.1.5.