Monoidal category action

In algebra, an action of a monoidal category ${\displaystyle (S,\otimes ,e)}$ on a category X is a functor

${\displaystyle \cdot :S\times X\to X}$

such that there are natural isomorphisms ${\displaystyle s\cdot (t\cdot x)\simeq (s\otimes t)\cdot x}$ and ${\displaystyle e\cdot x\simeq x}$, which satisfy the coherence conditions analogous to those in S.^[1] S is said to act on X.

Any monoidal category S is a monoid object in Cat with the monoidal product being the category product. This means that X equipped with an S-action is exactly a module over a monoid in Cat.

For example, S acts on itself via the monoid operation ⊗.