Proposition 15.3.6. The forgetful functor \[ U\colon \Cat _{\infty }^{\otimes ,\cart }\to \Cat _{\infty }^{\mathrm {prod}} \] is an equivalence, with inverse given by the assignment \(C\mapsto (C,\times )\).
Proof. Passing to opposites identifies \(\Cat _{\infty }^{\otimes ,\cart }\) with \(\Cat _{\infty }^{\otimes ,\cocart }\), and identifies the forgetful functor \(U\) with the corresponding forgetful functor \[ \Cat _{\infty }^{\otimes ,\cocart }\to \Cat _{\infty }^{\mathrm {coprod}}. \] The latter is an equivalence by Corollary 15.2.21: the functor \(C\mapsto (C,\amalg )\) is an equivalence whose composite with the forgetful functor is the identity on \(\Cat _{\infty }^{\mathrm {coprod}}\). Hence \(U\) is an equivalence as well.
By Proposition 15.3.5, the assignment \(C\mapsto (C,\times )\) lands in \(\Cat _{\infty }^{\otimes ,\cart }\). Full faithfulness of \(U\) uniquely lifts every finite-product-preserving functor \(C\to D\) to a symmetric monoidal functor \((C,\times )\to (D,\times )\). These lifts define a functor whose composite with \(U\) is the identity, and hence an inverse to \(U\). □
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