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Let $C$ be a $2$-category. There are two ways of dualizing $C$: The first one is well-known and also generalizes to arbitrary $(\infty,1)$-categories: We dualize "at each stage". The second one only dualizes the $2$-morphisms: We keep the underlying $1$-category, however we define $2$-morphisms $f \to g$ between $1$-morphisms $f,g : a \to b$ to be $2$-morphisms $g \to f$ in our given $C$. Thus we get a $2$-category $C'$. More abstractly, we may view $C$ as a category enriched over $\mathsf{Cat}$ and apply the usual dualization functor $\mathsf{Cat} \to \mathsf{Cat}$ to optain another category enriched over $\mathsf{Cat}$, namely $C'$.

Question. Is this kind of dualization well-known? Does $C'$ have a name? Is there a common notation? How do we call $2$-functors $D \to C'$ (which are roughly functors $C \to D$ which are covariant on $1$-morphisms and contravariant on $2$-morphisms)?

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There are three ways of dualising a 2-category $\mathfrak{K}$.

  • $\mathfrak{K}^\textrm{op}$ is the 2-category with the same 0-cells and 2-cells, but with 1-cells reversed. This is basically the enriched version of taking opposite categories.

  • $\mathfrak{K}^\textrm{co}$ is the 2-category with the same 0-cells and 1-cells, but with 2-cells reversed. An important example of this is the operation of forming the opposite of a 1-category: that is a 2-functor $\mathfrak{Cat}^\textrm{co} \to \mathfrak{Cat}$.

  • Of course, we can also do both: $\mathfrak{K}^\textrm{coop} = (\mathfrak{K}^\textrm{co})^\textrm{op}$.

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