I was reviewing some topics in Category Theory when I came across "monoid categories". I mean those with a single object $\{*\}$ and with the composition rule given by the product in some monoid. Let's call this category $M$.
In that case, if we fix some category $\mathscr{C}$, how can we understand the category $Fun(M,\mathscr{C})$? In $\mathscr{C}$ we have objects with a particular structure in the Hom set, is this notion useful?
In that case, I'd love to see some examples.
A functor from a group to a category maps the single object of the group category to some object of the target category. It maps the morphisms of the group category to morphisms of this object to itself.
Therefore a morphism from a group category to some other category is precisely the same as an action of the group on some object of that category.