Day convolution internal hom

Remark 16.1.2. Let \(\Oo \) be an \(\infty \)-operad for which \(\oDay (\Oo ,\Pp )\) exists for all \(\Pp \). Then the functor \(- \times \Oo \colon \Op _{\infty } \to \Op _{\infty }\) admits a right adjoint \[ \oDay (\Oo ,-)\colon \Op _{\infty } \to \Op _{\infty }. \]

Whenever the relevant Day convolution operads exist, this internal-hom description is functorial in both variables. An operad map \(u\colon \Oo '\to \Oo \) induces by precomposition an operad map \[ u^*\colon \oDay (\Oo ,\Pp )\longrightarrow \oDay (\Oo ',\Pp ), \] and an operad map \(v\colon \Pp \to \Pp '\) induces by postcomposition an operad map \[ v_*\colon \oDay (\Oo ,\Pp )\longrightarrow \oDay (\Oo ,\Pp '). \] The first map is adjoint to evaluation after precomposition with \(\id \times u\), while the second is adjoint to postcomposition of evaluation with \(v\). Their functoriality follows from the universal property. The same universal property gives a natural exponential law \[ \oDay \bigl (\Oo ,\oDay (\Pp ,\Qq )\bigr ) \simeq \oDay (\Oo \times \Pp ,\Qq ). \] Indeed, maps from an operad \(\Rr \) into either side are naturally equivalent to maps \(\Rr \times \Oo \times \Pp \to \Qq \).