Corollary 16.2.8. Let \(C\) and \(D\) be symmetric monoidal \(\infty \)-categories. Assume that \(C\) is small, that \(D\) is cocomplete (i.e., admits all small colimits), and that the tensor product of \(D\) preserves small colimits in each variable. Then the Day convolution operad \(\oDay (\Mm _C,\Mm _D)\) defines a symmetric monoidal structure \(\otimes _{\Day }\) on \(\Fun (C,D)\).
Proof. Every relative slice category \((\bigotimes _C^n)_{/c}\) is a pullback of the diagram \(C^n \to C \times C \leftarrow \Ar (C)\). Since \(C\) is small, so are \(C^n\) and \(\Ar (C)\), and hence so is \((\bigotimes _C^n)_{/c}\). The collection \(\Kk \) of Proposition 16.2.7 therefore consists of small \(\infty \)-categories, so both of its conditions are implied by the assumptions on \(D\). □
Generated from the authoritative LaTeX source.