Examples of 2-categories with multiple interesting proarrow equipment structures Proarrow equipments (also known as framed bicategories) are identity-on-objects locally fully faithful pseudofunctors $({-})_* \colon \mathcal K \to \mathcal M$ for which every 1-cell $f_*$ in the image has a right adjoint.
Though there is no reason in general that, for a given 2-category $\mathcal K$, there should be a canonical proarrow equipment structure, in many cases of interest, this is the case. For instance, proarrow equipments $({-})_* \colon \mathcal K \to \mathcal M$ satisfying a certain exactness property (Axiom C of Rosebrugh–Wood's Proarrows and cofibrations) are determined by the codiscrete cofibrations in $\mathcal K$. In practice, there often seems to be an evident choice of $\mathcal M$ and $({-})_*$.
What are some examples of 2-categories $\mathcal K$ for which there are distinct interesting proarrow equipment structures $({-})_* \colon \mathcal K \to \mathcal M$?
 A: I'm not sure whether this counts as "interesting", but if $(-)_* : \mathcal{K} \to \mathcal{M}$ is any proarrow equipment and $i:\mathcal{K}' \to \mathcal{K}$ is any locally fully faithful pseudofunctor, then the composite $\mathcal{K}'\to \mathcal{M}$ is again a proarrow equipment.
For instance, $(-)_*$ could be the proarrow equipment of $V$-enriched categories and profunctors, and $i$ could be the 2-functor $W\text{-Cat} \to V\text{-Cat}$ induced by some monoidal functor $W\to V$.  Such a functor isn't always locally fully faithful, but there are cases where it is, such as the "discrete objects" inclusions $\rm Set\to Cat$ or $\rm Set \to Top$.
A: Here is a different sort of non-answer: the category $\rm Dbl$ of (strict) double categories can be enhanced to a virtual equipment in two different ways, neither of which is canonical: using "horizontal" or "vertical" double profunctors respectively.  Indeed the lack of canonicity can be made precise, as the transposition automorphism of $\rm Dbl$ interchanges the two equipments.
This fails to answer the question not only because the equipments are only virtual, but because we are here talking only about the 1-category $\rm Dbl$.  This 1-category can be enhanced to a 2-category in two different ways, which correspond exactly to the two virtual equipments, in the usual codiscrete-cofibrations way.
