The usual Grothendieck construction is for pseudofunctors $I^{op}\to Cat$, where $I$ is a 1-category and $Cat$ is the 2-category of 1-categories. The Grothendieck construction produces a category with a functor into $I$, and the essential image is the Grothendieck fibrations over $I$.


My question: if we allow $I$ to be a bicategory, consider still $I^{op}\to Cat$. Is there a Grothendieck construction that produces a bicategory (?) with a functor into $I$? And what is the essential image of this construction (if there is one)? A quick candidate is the lax colimit of this functor, but I do not known if it is useful.


Added. We could go one step lower. Consider functor $I^{op}\to Set$, where $I$ is a 1-category. This is the same as an action of $I$. If I am not wrong, then the Grothendieck construction for $ I^{op}\to Set\to Cat$ will produces the action (translation) category.



A question on the Grothendieck construction
The bicategory of elements of a Cat-valued functor is defined in e.g. Street's *Fibrations in bicategories*; it's the same as the usual one, with 2-cells as described [here](http://ncatlab.org/nlab/show/2-limit#2ColimitsInCat). Its property of classifying lax transformations is also mentioned on that page ([this page](http://ncatlab.org/nlab/show/Grothendieck+construction#AsALaxColimit) describes that property in the usual setting; it is not hard to generalize it).


Baković's preprint *Fibrations of bicategories* ([PDF here](http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf)) goes into lots more detail on the Grothendieck construction for bicategory-valued functors. I don't know if it characterizes the fibrations that arise from category-valued functors (a quick text search suggests not), but one plausible guess is that they should be the ones with locally discrete fibres.


