Lemma 14.5.2. Let \((C, \otimes _C, \unit _C)\) be a symmetric monoidal \(\infty \)-category. If \(D \subseteq C\) is a full subcategory whose collection of objects \(D^{\simeq } \subseteq C^{\simeq }\) contains \(\unit _C\) and is closed under \(\otimes _C\), then \(D\) inherits a symmetric monoidal structure \((D, \otimes _D, \unit _D)\) from \(C\) such that the inclusion \(i\colon D \hookrightarrow C\) is a (strong) symmetric monoidal functor.
Proof. The previous construction provides a suboperad \(D^{\otimes } \subseteq C^{\otimes }\) and it remains to show that the functor \(D^{\otimes } \hookrightarrow C^{\otimes } \xrightarrow {p} \Span (\Fin )\) is still a cocartesian fibration. By full faithfulness of the inclusion, it will suffice to show that the \(p\)-cocartesian morphisms in \(C^{\otimes }\) starting in an object of \(D^{\otimes }\) have their targets in \(D^{\otimes }\). Using the inert-active factorization and Observation 14.2.7, we may reduce this to cocartesian morphisms of the form \[ (x_1,\dots ,x_n)\longrightarrow x_1\otimes \dots \otimes x_n. \] The claim is clear for \(n=1\). For \(n=0\) it is the condition that \(\unit _C\in D\), and for \(n=2\) it is the condition that \(x\otimes y\in D\) whenever \(x,y\in D\). The case \(n\geq 3\) reduces to \(n=2\) by induction, using that composites of cocartesian morphisms are again cocartesian. □
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