O monoidal Day convolution

Remark 18.3.4. A relative version of this result was proved by Lurie (2017), Theorem 2.2.6.2: if \(f\colon \Oo ' \to \Pp \) is a cocartesian fibration of \(\infty \)-operads, then the pullback functor \(f^*\colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Oo '}\) admits a right adjoint \[ \Nm _{\Oo '/\Pp }\colon (\Op _{\infty })_{/\Oo '} \to (\Op _{\infty })_{/\Pp }. \] By precomposing this functor with \(f^*\), it then follows that the product functor \(\Oo ' \times _{\Pp } - \colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Pp }\) admits a right adjoint \[ \oDay _{/\Pp }(\Oo ', -) \colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Pp }. \] If \(\Pp = \Comm \) is the terminal \(\infty \)-operad and \(\Oo ' \simeq \Mm _C\) for some \(C\), this specializes to Theorem 18.3.3.