Remark 4.6.4.2. Proposition 4.6.4.1 has an interpretation in the language of model categories. Let us regard the category $\operatorname{Set_{\Delta }}$ of simplicial sets as equipped with the Joyal model structure of Remark . Conditions $(2)$ and $(3)$ of Proposition 4.6.4.1 are equivalent to the requirement that the morphisms $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ are homotopic with respect to the Joyal model structure (in the sense of Definition ). Proposition 4.6.4.1 asserts that this is equivalent to the requirement that $f_0$ and $f_1$ are naturally isomorphic (in the sense of Definition 4.4.4.1).

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