Lemma 14.5.6. Let \(C\) be a symmetric monoidal \(\infty \)-category which admits internal homs in the sense that the functor \(- \otimes x \colon C \to C\) admits a right adjoint \(\iHom _C(x,-)\colon C \to C\) for every object \(x\) of \(C\). Let \(D\) be a full subcategory of \(C\) such that the inclusion \(D \hookrightarrow C\) admits a left adjoint, and assume that for objects \(x \in D\) and \(z \in C\) the internal hom \(\iHom _C(z,x)\) is again in \(D\). Then \(D\) obtains a symmetric monoidal structure such that the localization functor \(L \colon C \to D\) is symmetric monoidal.
Proof. We must prove the condition from Proposition 14.5.3. Let \(f\colon x \to y\) be an \(L\)-equivalence and let \(z \in C\) be an arbitrary object. We must show that the morphism \(L(f \otimes \id _z)\colon L(x \otimes z) \to L(y \otimes z)\) is invertible in \(D\). By the Yoneda lemma, this amounts to showing that for every object \(w \in D\) the map \[ (f \otimes \id _z)^*\colon \Hom _C(y \otimes z, w) \to \Hom _C(x \otimes z, w) \] is an equivalence of animae. Applying the adjunction \(- \otimes z \dashv \iHom _C(z,-)\), this map can be identified with the map \[ f^*\colon \Hom _C(y, \iHom _C(z,w)) \to \Hom _C(x, \iHom _C(z,w)). \] Since \(\iHom _C(z,w) \in D\) by assumption, this map is an equivalence due to the fact that \(f\) is an \(L\)-equivalence. □
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