Kerodon

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Example 5.5.4.9 ($\operatorname{\mathcal{S}}$-Valued Functors). Let $\operatorname{\mathcal{S}}$ denote the $\infty $-category of spaces (Construction 5.4.1.1), which we view as a full simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Set_{\Delta }})$, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a morphism of simplicial sets. Then the simplicial set $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ fits into pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \ar [r] \ar [d]^-{\pi } & \operatorname{\mathcal{S}}_{\ast } \ar [d] \\ \operatorname{\mathcal{C}}\ar [r]^-{\mathscr {F}} & \operatorname{\mathcal{S}}, } \]

where $\operatorname{\mathcal{S}}_{\ast }$ is the $\infty $-category of pointed spaces (Construction 5.4.3.1). In this case, Proposition 5.4.3.2 guarantees that the projection map $\pi : \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets.