In the 2-category **Cat** of small categories, for each category $C$ (an object of **Cat**) there is also the dual category (I dare not write "dual object") $C^{op}$.


Is ${op}$ the instance in **Cat** of a more general concept for 2-categories? (In the same sense that monads in **Cat** are a special case of monads-in-a-2-category $C$).



opposite category
Note that "taking the opposite" is an operation not just on categories, but also on functors and on natural transformations. Given a functor $F: C \to D$, there is an associated functor $F^{op}: C^{op} \to D^{op}$. (Notice that $(-)^{op}$ preserves the direction -- is covariant -- on functors.) Given a natural transformation $\eta: F \to G$ for functors $F, G: C \to D$, with a component $\eta X: F X \to G X$ for each object $X$ of $C$, there is an associated natural transformation $\eta^{op}: G^{op} \to F^{op}$. Note the change of direction: the components $\eta^{op} X$ go in the direction $G X \to F X$. 


In the lingo, there are 2 levels of duality for 2-categories $\mathbf{C}$, generating a Viergruppe so to speak. Given $\mathbf{C}$, we can either leave the directions of 1-cells and 2-cells unchanged (the identity), or we can 


* Reverse the directions of 1-cells (i.e., interchange their domains and codomains), but preserve the directions of 2-cells. This gives a 2-category denoted $\mathbf{C}^{op}$.
* Preserve the directions of 1-cells, but reverse the directions of 2-cells (i.e., interchange *their* domains and codomains, which are 1-cells). This gives a 2-category denoted $\mathbf{C}^{co}$.
* Reverse directions of both 1-cells and 2-cells, i.e., compose the preceding two operations (in either order; the result is the same). This gives a 2-category denoted $\mathbf{C}^{coop}$.


The "op" operation on $\mathbf{Cat}$ gives then a 2-functor $\mathbf{Cat}^{co} \to \mathbf{Cat}$. (This could be a little confusing, I suppose, since the "op" operation doesn't match the "co", but it's too late to change it now.) 


Not all 2-categories $\mathbf{C}$ admit such an operator $(-)^\ast: \mathbf{C}^{co} \to \mathbf{C}$. Moreover, the operator $(-)^{op}$ on $\mathbf{Cat}$ is dual to itself in an obvious sense. Formally, given any 2-functor $F: \mathbf{C} \to \mathbf{D}$, one can form another 2-functor $F^{co}: \mathbf{C}^{co} \to \mathbf{D}^{co}$; in particular, this could be applied to the "op" operation on $\mathbf{Cat}$, giving 


$$op^{co}: \mathbf{Cat} \to \mathbf{Cat}^{co}$$ 


and the point now is that $op$ is inverse to $op^{co}$ in a suitable sense. So "op" exhibits $\mathbf{Cat}$ as being co-self-dual. So co-self-dual 2-categories is the operative concept; I'm actually not sure how standard this name is though. 


