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 $C$).
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.
(Below by "$2$-category" I mean "bicategory.")
It's more fun to think about opposites, not in the $2$-category of categories, functors, and natural transformations, but in the $2$-category of categories, bimodules / distributors / profunctors, and natural transformations of bimodules. I prefer the bimodule terminology so let me use that. Recall that a $(C, D)$-bimodule, where $C, D$ are categories, is a functor $M : C^{op} \times D \to \text{Set}$. These naturally form a category. There is a tensor product operation taking a $(C, D)$-bimodule and a $(D, E)$-bimodule and returning a $(C, E)$-bimodule which gives the composition in this $2$-category. Moreover, the product of categories gives this $2$-category a symmetric monoidal structure.
The punchline is now that $C^{op}$ is the monoidal dual of $C$ in this symmetric monoidal $2$-category. The evaluation map is given by
$$\text{Hom}(-, -) : (C^{op} \times C) \times 1 \to \text{Set}$$
(regarded as a $(C^{op} \times C, 1)$-bimodule) and the coevaluation map is given by
$$\text{Hom}(-, -) : 1 \times (C \times C^{op}) \to \text{Set}$$
(regarded as a $(1, C \times C^{op})$-bimodule, and I need to switch the inputs here). In particular, every object of this $2$-category is dualizable.
The same story works with essentially no modification if we enrich everything in a reasonable symmetric monoidal category $V$. In particular, when $V = \text{Ab}$ we get a $2$-category into which the usual bimodule $2$-category of rings, bimodules, and morphisms of bimodules embeds, and we recover the observation that every ring $R$ is dualizable in this $2$-category with monoidal dual $R^{op}$. This is one reason bimodule categories are an attractive target category for TQFTs.
To make a link between Todd's answer and Qiaochu's answer: if $\mathbb{W}$ is a 2-category, then a functor:
$$(-)^* \colon \mathbb{W}^{co} \rightarrow \mathbb{W}$$
is called a "duality involution" if it is self-inverse and (pseudo) naturally satisfies:
$$\mathit{DFib}(A\times B, C) \approx \mathit{DFib}(A, B^* \times C)$$ where $\mathit{DFib}(X, Y)$ is the category of discrete fibrations from $X$ to $Y$ in $\mathbb{W}$. Observe that in case $\mathbb{W} = \mathbf{Cat}$ discrete fibrations are equivalent to profunctors by the generalised Grothendieck construction and its inverse; the above equivalence is then induced by pairing (evaluation) $B \times B^* \nrightarrow 1$ from autonomous (weak) 2-category of profunctors.
This concept of "duality involution" was a crucial part of the definition of a 2-topos given by Mark Weber in "Yoneda structures from 2-toposes".
