Day convolution on slices

Let \(\square_X\) denote Day convolution in \(\Ar(T_{/X})\), with respect to the minimum operation on \([1]\), where \(\Ar(T_{/X})=\Fun([1],T_{/X})\). Given maps \(u\colon A \to B\) and \(v\colon C \to D\) in \(T_{/X}\), the map \(u \square_X v\) is the dashed arrow in the pushout square:

This operation is also known as the pushout product of \(u\) and \(v\) over \(X\).