Corollary 14.4.10. Let \(C\) be a monoidal \(\infty \)-category and let \(I\) be a small \(\infty \)-category such that \(C\) admits \(I\)-indexed limits. Then \(\Alg (C)\) admits \(I\)-indexed limits, and the forgetful functor \(\Alg (C)\to C\) creates and preserves them. The same holds for the forgetful functor \(\CAlg (C)\to C\) when \(C\) is symmetric monoidal.
Proof. This is the one-color case of Proposition 14.4.9: a monoidal \(\infty \)-category is an \(\Assoc \)-monoidal \(\infty \)-category, with \(\Alg (C)=\Alg _{\Assoc /\Assoc }(C)\), and a symmetric monoidal \(\infty \)-category is a \(\Comm \)-monoidal \(\infty \)-category, with \(\CAlg (C)=\Alg _{\Comm /\Comm }(C)\). □
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