Kerodon

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Proposition 7.5.3.4. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty$-categories, and let $\alpha : \mathscr {F} \rightarrow \mathscr {F}^{+}$ be the natural transformation of Construction 7.5.3.3. Then $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram, and $\alpha$ is a levelwise categorical equivalence. Moreover, $\alpha$ is also a monomorphism.

Proof. It follows from Corollary 7.5.3.2 that the diagram $\mathscr {F}^{+}$ is isofibrant. To see that $\alpha$ is a monomorphism, we observe that for each object $C \in \operatorname{\mathcal{C}}$, the functor

$\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {F}^{+}(C) = \operatorname{Fun}^{ \operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ }), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}))$

has a left inverse, given by the evaluation map

$\operatorname{ev}_{C}: \operatorname{Fun}^{ \operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ }), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \mathscr {F}(C).$

Since $\operatorname{ev}_{C}$ is a trivial Kan fibration (Proposition 5.3.1.7), it follows that $\alpha _{C}$ is an equivalence of $\infty$-categories. $\square$