Description Day convolution

Remark 16.2.9. The proof of the proposition provides an explicit description of the tensor product functor \[ \bigotimes _{\Day }^I\colon \Fun (C,D)^I \to \Fun (C,D) \] for every finite set \(I\). For \(I = \lra {0}\) and \(I = \lra {2}\), this specializes as follows:

• The monoidal unit \(\unit _{\Day }\colon C \to D\) of the Day convolution monoidal structure is the left Kan extension of \(\unit _D\colon * \to D\) along \(\unit _C\colon * \to C\).
• The Day convolution \(F \otimes _{\Day } G\) of two functors \(F,G\colon C \to D\) is the left Kan extension of \(C \times C \xrightarrow {F \times G} D \times D \xrightarrow {\otimes _D} D\) along \(\otimes _C\colon C \times C \to C\).