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.