Proposition 18.2.3 (Multiplicative Yoneda embedding). Let \(\Oo \) be a small \(\infty \)-operad and put \(D := \Env (\Oo )\). The composite \[ \Oo \xrightarrow {\eta _{\Oo }} \Mm _D \xrightarrow {\Mm _Y} \Mm _{\PSh (D)} \] is fully faithful in the sense that it induces equivalences on all multimorphism animae. Moreover, if \(\Oo \) admits operadic limits indexed by some \(\infty \)-category \(I\), then the induced functor on colors \[ \Oo _{\lra {1}} \to \PSh (D) \] preserves \(I\)-indexed limits.
Proof. Let \(x_1,\dots ,x_n,y\) be colors of \(\Oo \). Since \(Y\) is strong symmetric monoidal by Proposition 18.2.2, we have natural equivalences \begin {align*} \Mm _{\PSh (D)}(Yx_1,\dots ,Yx_n;Yy) &\simeq \Hom _{\PSh (D)}(Yx_1 \otimes ^{\Day } \dots \otimes ^{\Day } Yx_n,Yy) \\ &\simeq \Hom _{\PSh (D)}(Y(x_1 \otimes \dots \otimes x_n),Yy) \\ &\simeq \Hom _D(x_1 \otimes \dots \otimes x_n,y) \\ &\simeq \Oo (x_1,\dots ,x_n;y). \end {align*}
The first and last equivalences are induced by the unit \(\eta _{\Oo }\) and agree with the identifications of Lemma 17.3.17. This proves full faithfulness. For the final claim, let \(y \to F\) be an operadic limit cone in \(\Oo _{\lra {1}}\). By Remark 18.1.2, its image in \(D = \Env (\Oo )\) is an ordinary limit cone. Since the Yoneda embedding preserves limits, its image in \(\PSh (D)\) is again a limit cone. □
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