Multiplicative Yoneda image closed under limits

Corollary 18.2.4. Let \(\Oo \) be a small \(\infty \)-operad that admits finite operadic limits and put \(D := \Env (\Oo )\). Then the color functor \[ \Oo _{\lra {1}} \to \PSh (D) \] induced by the multiplicative Yoneda embedding is fully faithful, preserves finite limits, and has essential image closed under finite limits in \(\PSh (D)\). In particular, by Lemma 14.1.8, we may identify \(\Oo \) with the full suboperad of \(\Mm _{\PSh (D)}\) spanned by this essential image. □