Proposition 14.4.9 (cf. [Lurie (2017), Corollary 3.2.2.5]). Let \(q\colon \Pp \to \Oo \) be a morphism of \(\infty \)-operads, let \(C\) be an \(\Oo \)-monoidal \(\infty \)-category, and let \(I\) be a small \(\infty \)-category. Assume that each fiber \(C_x\), for \(x\in \Oo ^{\simeq }\), admits \(I\)-indexed limits. Then \(\Alg _{\Pp /\Oo }(C)\) admits \(I\)-indexed limits, and the evaluation functors \[ \ev _y\colon \Alg _{\Pp /\Oo }(C)\to C_{q(y)}, \qquad A\mapsto A_y \qquad (y\in \Pp ^{\simeq }), \] jointly create them.
Proof. The proof is analogous to that of Proposition 14.3.10. Applying \(\Fun (I,-)\) fiberwise gives an \(\Oo \)-monoidal category whose fiber at \(x\) is \(\Fun (I,C_x)\), and algebras in it identify with \(I\)-diagrams in \(\Alg _{\Pp /\Oo }(C)\). The fiberwise constant-diagram functors form an \(\Oo \)-monoidal functor, with lax \(\Oo \)-monoidal right adjoint given in each fiber by \(\lim _I\). By Proposition 14.4.7, these functors form a relative adjunction over \(\Oo ^{\otimes }\), so Lemma 14.3.5 gives the required right adjoint to the constant-diagram functor after applying \(\Fun _{/\Oo ^{\otimes }}(\Pp ^{\otimes },-)\) and restricting to operad maps. Its evaluation at \(y\) is \(\lim _I\) in \(C_{q(y)}\). The evaluation functors jointly detect isomorphisms, so they jointly create the limits. □
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