Given a monoidal category a (left or right) -actegory is a category together with a (left or right) coherent action of on . Depending on an author and context, the left coherent action of on is a morphism of monoidal categories in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict monoidal functor. Right coherent actions correspond to the monoidal functors into the category with the opposite tensor product.
-actegories, colax -equivariant functors and natural transformations of colax -equivariant functors form a strict 2-category . A monad in amounts to a pair of a monad in and a distributive law between the monad and an action of .
The notion of -action (hence a -actegory) is easily extendable to bicategories (see Baković’s thesis).
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M. Kelly, G. Janelidze, A note on actions of a monoidal category, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61–91 link
P. Schauenburg, Actions of monoidal categories and generalized Hopf smash products, J. Algebra 270 (2003) 521–563 (remark: actegories with action in the strong monoidal sense)