Monoidal category action
In algebra, an action of a monoidal category S on a category X is a functor

such that there are natural isomorphisms  and  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 ⊗.

References 

Monoidal categories
Functors