Theorem 18.3.5 (Multimorphism formula for Day convolution). Let \(C\) be a small symmetric monoidal \(\infty \)-category and let \(\Oo \) be an \(\infty \)-operad. For every finite set \(I\) and functors \(F_i,G\colon C \to \Oo _{\lra {1}}\), there is a natural equivalence \[ \oDay (\Mm _C,\Oo )(\{F_i\}_{i \in I};G) \simeq \int _{\{c_i\}_{i \in I}\in C^I} \Oo (\{F_i(c_i)\}_{i \in I};G(\bigotimes \nolimits ^I_C c_i)). \]
Proof. Put \(D := \Env (\Oo )\) and identify \(\Oo \) with the full suboperad of \(\Mm _D\) spanned by the singleton tuples, as in the proof of Theorem 18.3.3. By Lemma 18.3.2, the operad \(\oDay (\Mm _C,\Oo )\) is the full suboperad of \(\oDay (\Mm _C,\Mm _D)\) spanned by the \(\Oo _{\lra {1}}\)-valued functors. Therefore its multimorphism animae agree with the corresponding multimorphism animae in \(\oDay (\Mm _C,\Mm _D)\). By Proposition 16.2.6, we get \[ \oDay (\Mm _C,\Oo )(\{F_i\}_{i \in I};G) \simeq \Nat \left (\bigotimes \nolimits ^I_D\circ \prod _{i \in I}F_i,G\circ \bigotimes \nolimits ^I_C\right ). \] Using the end formula for natural transformations from Proposition 23.6.6, the right-hand side is equivalent to \[ \int _{\{c_i\}_{i \in I}\in C^I} \Hom _D\left (\bigotimes \nolimits ^I_D F_i(c_i),G(\bigotimes \nolimits ^I_C c_i)\right ). \] By Lemma 17.3.17, the integrand is naturally equivalent to \[ \Oo (\{F_i(c_i)\}_{i \in I};G(\bigotimes \nolimits ^I_C c_i)). \] Taking ends preserves this equivalence of integrands, giving the desired formula. □
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