Lemma 18.3.1. Let \(i\colon \Oo \hookrightarrow \Qq \) be a full suboperad. For every \(\infty \)-operad \(\Rr \), the induced functor \[ \Fun _{\Op _{\infty }}(\Rr ,\Oo ) \to \Fun _{\Op _{\infty }}(\Rr ,\Qq ) \] is fully faithful. Its essential image consists of those operad maps \(\Rr \to \Qq \) whose colors land in \(\Oo _{\lra {1}} \subseteq \Qq _{\lra {1}}\).
Proof. This is clear from the fact that the total category \(\Oo ^{\otimes }\) is the full subcategory of \(\Qq ^{\otimes }\) spanned by those finite tuples whose entries are colors of \(\Oo \). □
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