Corollary 19.4.7 ([Lurie (2017), Corollary 4.7.1.40]). Let \(D\) be left-tensored over a monoidal \(\infty \)-category \(C\), and let \(M \in D\) admit an endomorphism object \(\End (M) \in C\). Then \(\End (M)\) admits a preferred structure of an associative algebra, and \(M\) admits a preferred structure of a left module over \(\End (M)\). Moreover, there is an equivalence of \(\infty \)-categories making the following diagram commute:
Proof. By definition, \(\End (M)\) is terminal in \(C[M]\). The monoidal structure from Theorem 19.4.6 and Corollary 14.4.11 therefore give \(\End (M)\) a unique associative algebra structure in \(C[M]\). Its image in \(C\) is an associative algebra, while its image under the equivalence of Theorem 19.4.6 is a left module structure on \(M\).
The forgetful functor \(\Alg (C[M]) \to \Alg (C)\) is a right fibration by [Lurie (2017), Proposition 4.7.1.39]. Since its total category has the terminal object just constructed, this right fibration is represented by \(\End (M)\). This gives the displayed equivalence. □
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