I'm not sure if this is the right place to ask this question, but I'll ask it anyway, in the hope that some kindly Australian (true or honorary) is passing by and takes pity on me...


In *Fibrations in bicategories*, Street shows that V-profunctors are exactly the codiscrete cofibrations in the 2-category V-Cat (i.e. the discrete 2-sided fibrations in the opposite of V-Cat). Rosebrugh and Wood later generalized this to well-behaved proarrow equipments.


When V=Set, so that V-Cat is Cat, then, codiscrete cofibrations turn out to be essentially the same thing as discrete fibrations. My question is



> 
> Why is this true? That is, for which bicategories K is DFib(B,A) equivalent to CodCofib(B,A) for all objects A and B?
> 
> 
> 


I ask because (aside from curiosity) I'd like to know whether I can expect a 'biprofunctor' $L^{\mathrm{op}} \times K \to \mathrm{Cat}$ to be the same as a discrete fibration in Bicat, or even whether this is true in the strictly Cat-enriched case. In my specific case L and K are locally discrete, if that makes a difference.



Comparing discrete fibrations and their duals
The fact that codiscrete cofibrations and discrete fibrations are equivalent is a very special exactness property of the 2-category Cat. I can't, off the top of my head, think of any other interesting examples. It's not true, for example in the 2-category 2-Cat of 2-categories, 2-functors, and 2-natural transformations you refer to, or in most (all?) other 2-categories of the form V-Cat. Consider, for example, the case where V=Cat and A=B=1. Then CodCofib(1,1) is just (the underlying ordinary category of) Cat. On the other hand, Fib(1,1) is 2-Cat, but
discreteness of a fibration says that the 2-category A has an underlying ordinary category which is discrete. 


If you want to move from 2-Cat to Bicat, you'd first have to decide how you want to make Bicat
into a 2-category (or decide what you mean by internal discrete fibrations or codiscrete cofibrations in a tricategory). But I wouldn't hold out too much hope ....


