Theorem 18.3.3. Let \(C\) be a symmetric monoidal \(\infty \)-category. For every \(\infty \)-operad \(\Oo \), the Day convolution operad \(\oDay (\Mm _C,\Oo )\) exists.
Proof. Put \(D := \Env (\Oo )\). By Lemma 17.3.17, the unit \[ \eta _{\Oo }\colon \Oo \to \Mm _D \] is fully faithful: its essential image is the full suboperad spanned by the singleton tuples of colors in \(D\). We may therefore identify \(\Oo \) with this full suboperad by Lemma 14.1.8. The Day convolution operad \(\oDay (\Mm _C,\Mm _D)\) exists by Theorem 16.2.4, so Lemma 18.3.2 gives the result. □
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