category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
As monoidal categories are a vertical categorification of monoids, actegories are a vertical categorification of actions of a monoid. So given a monoidal category $(C,\otimes,I,l,r,a)$ an actegory is another category $D$ with a notion of “tensor by object of $C$”, i.e., a functor:
that is associative and unital up to natural isomorphism with respect to $\otimes$ in ways that generalize actions of a monoid, and satisfy coherence laws similar to those of a monoidal category.
For any category $A$, the category of endofunctors $End(A)$ is monoidal with respect to the (horizontal) composition (the composition of functors and the Godement product for natural transformations).
Given a monoidal category $(C,\otimes,I,l,r,a)$ a (left or right) $C$-actegory is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on an author and context, the left coherent action of $C$ on $A$ is a morphism of monoidal categories $C\to End(A)$ 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 $End(A)$ with the opposite tensor product.
$C$-actegories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a strict 2-category $_C Act^c$. A monad in $_C Act^c$ amounts to a pair of a monad in $Cat$ and a distributive law between the monad and an action of $C$.
The notion of $C$-action (hence a $C$-actegory) is easily extendable to bicategories (see Baković‘s thesis).
A (left) $\mathcal{C}$-(pseudo)actegory is
satisfying a pentagonal and two triangular laws (see KJ01, diagg. (1.1)-(1.3)) that witness the coherence of $\lambda$ and $\alpha$ with the unitors and associators of $\mathcal{C}$.
If a category $D$ is enriched in $C$ with copowers, then the copower structure forms an actegory on the ordinary category underlying $D$.
Conversely, if an actegory is such that the functor $(-)\oslash d:C\to D$ has a right adjoint for all objects $d$ of $D$, then the right adjoints $D(d,-):D\to C$ provide an enrichment of $D$ in $C$ for which the action is a copower. See KJ01.
Bodo Pareigis, Non-additive ring and module theory I. General theory of monoids, Publ. Math. Debrecen 24 (1977), 189–204. MR 56:8656; Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms, Publ. Math. Debrecen 24 (351–361) 1977.
Max Kelly, George 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)
Zoran Škoda, Distributive laws for actions of monoidal categories, arxiv:0406310, Equivariant monads and equivariant lifts versus a 2-category of distributive laws, arxiv:0707.1609