It's well known that we can exhibit the comma category as a particular type of 2-limit in Cat. When working with 2-categories, there is a naïve comma object given by by boosting up the ordinary diagram for a comma category, but this fails to have many of the useful formal properties we might expect. 


The answer is that we should consider appropriate 2-subcategories of the 2-comma category, the Grothendieck fibrations (or opposite Grothendieck fibrations depending on the application). However, at least as far as I can tell, these 2-categories $Cat\downarrow\_{Fib} C$ and $Cat\downarrow\_{OpFib} C$ do not have obvious universal constructions. 


Is the situation hopeless, or is there a tricky way to realize these objects as 2 or 3-limits? 


Please note: I am aware that there is an equivalence between these guys and the relevant pseudofunctor categories, but that is not what I'm looking for. That would be, for my purposes, begging the question.



Can we exhibit the 2-category of Grothendieck fibrations as a 2 (or 3)-limit?
I'm not sure whether you're working in Cat or 2-Cat; I'll assume the latter.


If B is an object of a finitely (2-)complete 2-category K, then the 2-category Fib(B) of fibrations over B is monadic over K/B, where 'fibration' is meant in Street's sense (see [nLab](http://ncatlab.org/nlab/show/fibration+in+a+2-category) or Street's *Fibrations in bicategories* in the Cahiers or the paper cited by Buschi Sergio above). The monad takes $f \colon E \to B$ to the projection $B/f \to B$. So Fib(B) has the universal property of an [Eilenberg--Moore object](http://ncatlab.org/nlab/show/Eilenberg-Moore+category), i.e. it's a lax limit.


So if K is 2-Cat, then as long as your definition of fibrations matches this one, you have your universal property.


