Monoidal category action

In algebra, an action of a monoidal category S on a category X is a functor
 * $$\cdot: S \times X \to X$$

such that there are natural isomorphisms $$s \cdot (t \cdot x) \simeq (s \cdot t)\cdot x$$ and $$e \cdot x \simeq x$$ and those natural isomorphism satisfy the coherence conditions analogous to those in S. If there is such an action, S is said to act on X.

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