Lemma 18.3.2. Let \(i\colon \Oo \hookrightarrow \Qq \) be a full suboperad, let \(C\) be a symmetric monoidal \(\infty \)-category, and assume that \(\oDay (\Mm _C,\Qq )\) exists. Let \[ \Ww \subseteq \oDay (\Mm _C,\Qq ) \] be the full suboperad spanned by those functors \(C \to \Qq _{\lra {1}}\) whose values lie in \(\Oo _{\lra {1}}\). Then \(\Ww \) is a Day convolution operad \(\oDay (\Mm _C,\Oo )\).
Proof. The evaluation map for \(\oDay (\Mm _C,\Qq )\) restricts to a map \[ \Ww \times \Mm _C \to \Oo , \] because on colors it sends a pair \((F,c)\) to \(F(c)\). We verify the defining universal property. Let \(\Rr \) be an arbitrary \(\infty \)-operad. By Lemma 18.3.1, maps \(\Rr \to \Ww \) are precisely maps \(\Rr \to \oDay (\Mm _C,\Qq )\) whose color functors land in \(\Oo _{\lra {1}}\). Transposing along the Day convolution adjunction for \(\Qq \), these are precisely maps \[ \Rr \times \Mm _C \to \Qq \] whose colors land in \(\Oo _{\lra {1}}\). Applying Lemma 18.3.1 again, this is the same as giving a map \(\Rr \times \Mm _C \to \Oo \). This identification is natural in \(\Rr \), hence \(\Ww \) satisfies the universal property of \(\oDay (\Mm _C,\Oo )\). □
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