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.[1] If there is such an action, S is said to act on X.

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

References

Weibel, Ch. IV, Definition 4.7.

C. Weibel "The K-book: An introduction to algebraic K-theory"

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[1] Weibel, Ch. IV, Definition 4.7.