nLab
monoid object

A monoid object in a category $C$ is a monoid internal to $C$.

By the microcosm principle, in order to define monoid objects in $C$, $C$ itself must be a “categorified monoid” in some way. The natural requirement is that it be a monoidal category. In fact, it suffices if it is a multicategory.

Compare to a group object, which in contrast can only be defined using finite products, rather than an arbitrary monoidal structure on the category $C$.