Remark 1.3.6.6. Let $\operatorname{\mathcal{C}}$ be a category. Then the counit of the adjunction described in Corollary 1.3.6.5 induces an isomorphism of categories $\mathrm{h} \mathit{\operatorname{N}}_{\bullet }(\operatorname{\mathcal{C}}) \xrightarrow {\sim } \operatorname{\mathcal{C}}$ (this is a restatement of Proposition 1.3.3.1). In other words, every category $\operatorname{\mathcal{C}}$ can be recovered as the homotopy category of its nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
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