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Is this kind of functor $\mathsf{Set}/M×\mathsf{Set}/M\to \mathsf{Set}/M$, with $M$ a monoid...
In general, for any monoidal category $(\mathcal M, \otimes, I)$ and a monoid $(m, *, i) \in \mathcal M$, the slice category $\mathcal M/m$ inherits the monoidal structure pointwise, i.e. the tensor p …