Lemma 16.1.3. Assume \(\oDay (\Oo ,\Pp )\) exists. Then the underlying \(\infty \)-category of \(\oDay (\Oo ,\Pp )\) is the \(\infty \)-category of functors \(\Oo _{\lra {1}} \to \Pp _{\lra {1}}\) between the underlying \(\infty \)-categories of \(\Oo \) and \(\Pp \): \[ \oDay (\Oo ,\Pp )_{\lra {1}} \quad \simeq \quad \Fun (\Oo _{\lra {1}},\Pp _{\lra {1}}). \]
Proof. By Lemma 14.1.20, the universal property of Day convolution, and Lemma 14.1.23, we have natural equivalences \begin {align*} \oDay (\Oo ,\Pp )_{\lra {1}} &\simeq \Fun _{\Op _{\infty }}(\Triv ,\oDay (\Oo ,\Pp )) \\ &\simeq \Fun _{\Op _{\infty }}(\Triv \times \Oo ,\Pp ) \\ &\simeq \Fun _{\Op _{\infty }}(\Triv _{\Oo _{\lra {1}}},\Pp ) \\ &\simeq \Fun (\Oo _{\lra {1}},\Pp _{\lra {1}}), \end {align*}
where the final equivalence is another instance of Lemma 14.1.20. □
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