# Kerodon

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Remark 5.3.4.10 (Universality). Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty$-categories, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Applying Corollary 5.3.2.23, we obtain a bijection from the set of morphisms $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ in the category $(\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ to the set of natural transformations $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Unwinding the definitions, we see that $\alpha$ factors through the subfunctor $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} \subseteq \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ if and only if $\lambda$ satisfies condition $(1)$ of Definition 5.3.4.2. If this condition is satisfied, then $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is a levelwise categorical equivalence if and only if $\lambda$ satisfies condition $(2)$ of Definition 5.3.4.2. We therefore obtain a bijection

$\xymatrix@R =50pt@C=50pt{ \{ \textnormal{Levelwise categorical equivalences \alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}} \} \ar [d]^{\Phi } \\ \{ \textnormal{Scaffolds \lambda : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}} \} . }$

Concretely, this bijection carries a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ to the composite map

$\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow { \alpha } \underset { \longrightarrow }{\mathrm{holim}}( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ) \xrightarrow { \lambda _ u } \operatorname{\mathcal{E}},$

where $\lambda _{u}$ is the universal scaffold of Construction 5.3.4.7.