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 concept of "duality involution" was a crucial part of a definition of a 2-topos given by Mark Weber in "Yoneda structures from 2-toposes".

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".