Kerodon

Construction 8.6.5.6. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, let $\kappa $ be an uncountable cardinal, and let let $\operatorname{\mathcal{S}}^{< \kappa }$ denote the $\infty $-category of essentially $\kappa $-small spaces. We let $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ denote the relative exponential of Construction 4.5.9.1. By construction, we can identify vertices of $\operatorname{Fun}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ with pairs $(C, \mathscr {F}_{C} )$, where $C$ is a vertex of $\operatorname{\mathcal{C}}$ and $\mathscr {F}_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is a functor of $\infty $-categories. We let $\operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )$ denote the full simplicial subset of $\operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa })$ spanned by those vertices $(C, \mathscr {F}_{C} )$ where the functor $\mathscr {F}_{C}$ is corepresentable by an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C}$. In what follows, we will generally write $\pi : \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{ < \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ for the projection map, and $\pi ^{\operatorname{corep}}: \operatorname{Fun}^{\operatorname{corep}}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa }) \rightarrow \operatorname{\mathcal{C}}$ for the restriction of $\pi $ to the simplicial subset $\operatorname{Fun}^{\operatorname{corep}}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \subseteq \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{ < \kappa } )$.