Kleisli bi-categories

Kleisli bi-categories

[Reiko Heckel]

Dear Category Theorists,

I'm looking for a reference on the following bi-categorical variant of
the construction of Kleisli categories.

Start from a pseudo-free construction $F$ between (2-)categories $\cat
C$ and $\cat D$. In the case I'm interested in, $F$ arises from a finite
cocompletion. For example, $\cat C$ is the category of small categories
and $\cat D$ is the category of small categories with finite colimits.  

Constructing (in the usual way) the Kleisli category $\cat K$ for $F$
leads to a bi-category since the universal property used to define the
composition in $\cat K$ holds only up to a unique natural isomorphism.   

Then, there exists a (unique?) homomorphism of bi-categories $G: \cat K
\to \cat D$ which commutes with the obvious embedding of $\cat C$ into
$\cat K$ and (the pseudo functor induced by) $F$. 

I understand that pseudo-free constructions of the above kind can be
axiomatized by means of Kock-Zoeberlein (KZ) monads. Thus, any reference
on Kleisli-like constructions for KZ monads would do. 
For me, it was simpler to work directly with the universal properties,
in particular since all the coherence laws are automatic.

Any hints or references to relevant literature would be much
appreciated.

Best regards,
Reiko Heckel.

-- 
 Reiko Heckel                             E-Mail:reiko@uni-paderborn.de
 Univ. GH Paderborn, FB 17                Tel: ++49-05251-60-3356
 Warburger Str. 100, E4.130               Fax: ++49-05251-60-3431
 33098 Paderborn, Germany