The notion of a Grothendieck fibration in the 2-category $CAT$ of small categories, can be written down to make sense for any 2-category, and such a morphism in a 2-category is called a fibration:

http://ncatlab.org/nlab/show/fibration+in+a+2-category

Now, in the case of $CAT$, consider a category $C$, then I *believe* that the 2-category of Grothendieck fibrations over $C$ is a (non-full) coreflective 2-subcategory of the slice 2-category $CAT/C$ (where morphisms from $D \stackrel{f}{\rightarrow}C$ to $D' \stackrel{g}{\rightarrow}C$
are pairs $(h,\alpha)$, such that $\alpha:gh \Rightarrow f$). At least, this seems to be true if $C$ is a groupoid.

My argument is to describe explicitly the right-adjoint which sends a functor $f:D \to C$ to the Grothendieck construction of the presheaf of categories $c \mapsto Hom_{Cat/C}\left(C/c\stackrel{}{\searrow}C,D\stackrel{f}{\searrow}C\right).$

If this doesn't work in general, it should at least if $C$ is a groupoid.

Now, let $E$ be a 2-category (if need be, we can assume its a (2,1)-category). Is is true that for every $e$ in $E$, that the 2-category of fibrations over $e$ is coreflective in $E/e$?

P.S. If this is not true in general, when is it true?