Day convolution operad

Definition 16.1.1. Let \(\Oo \) and \(\Pp \) be \(\infty \)-operads. An \(\infty \)-operad \(\oDay (\Oo ,\Pp )\) is called a Day convolution operad for \(\Oo \) and \(\Pp \) if it comes equipped with a morphism of \(\infty \)-operads \[ \ev \colon \oDay (\Oo ,\Pp ) \times \Oo \to \Pp \] satisfying the following universal property: if \(\Qq \) is another \(\infty \)-operad, the composite \[ \hspace {-6pt} \Fun _{\Op _{\infty }}(\Qq ,\oDay (\Oo ,\Pp )) \xrightarrow {- \times \Oo } \Fun _{\Op _{\infty }}(\Qq \times \Oo , \oDay (\Oo ,\Pp ) \times \Oo ) \xrightarrow {\ev \circ -} \Fun _{\Op _{\infty }}(\Qq \times \Oo , \Pp ) \] is an equivalence. Informally, this means we have the following classification of operad morphisms into \(\oDay (\Oo ,\Pp )\): \[ \{\text {Operad morphisms $\Qq \to \oDay (\Oo ,\Pp )$}\} \quad \leftrightsquigarrow \quad \{\text {Operad morphisms $\Qq \times \Oo \to \Pp $}\}. \] Note that this property uniquely determines \(\oDay (\Oo ,\Pp )\) if it exists.