Definition 19.1.10. Let \(\Oo \) be a unital \(\infty \)-operad and let \(C\colon \Oo ^{\otimes } \to \Cat _{\infty }\) be an \(\Oo \)-monoidal \(\infty \)-category. For every color \(x \in \Oo ^{\simeq }\), the unique multimorphism \(e_x \in \Oo (\{\};x)\) defines a functor \[ \unit _x\colon * \simeq C(\{\}) \to C(\{x\}) = C_x. \] We refer to the corresponding object \(\unit _x \in C_x\) as the monoidal unit of \(C_x\).
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