Realization of an ∞-category via its nerve

Proof. The colimit-functor is by definition a left adjoint to the constant simplicial anima functor: \[ \abs {-}\colon \s \An \rightleftarrows \An \noloc \const . \] For any anima \(X\), the constant simplicial anima \(\const _X\colon \simp \catop \to \An \) is a complete Segal anima (equivalent to the nerve \(N(X)\) of \(X\)), and hence this adjunction restricts to an adjunction \[ \abs {-}\colon \mathrm {CS}(\An ) \rightleftarrows \An \noloc \const . \] Under the equivalence \(\Cat _{\infty } \simeq \mathrm {CS}(\An )\) from Theorem 24.1.10, the constant functor corresponds to the inclusion \(\An \hookrightarrow \Cat _{\infty }\), and by uniqueness of adjoints this implies the claim. □