ct.category theory - Does F_* have a description in terms of the Grothendieck construction?

I guess it depends what you would accept as a "description in terms of the Grothendieck construction."

For each $d in D$, we have the projection $pi_d : d/F rightarrow C$. Given some $gamma : C rightarrow Set$, I can form the composite $$ gamma circ pi_d : d/F rightarrow C rightarrow Set$$

This determines a discrete opfibration $int gamma circ pi_d$ over $d/F$. Let $Gamma_{d/F}$ denote the set of sections of the projection $int gamma circ pi_d rightarrow d/F$. Then I believe you will find that $F_* gamma$ is the Grothendieck construction on the functor $d mapsto Gamma_{d/F}$.

Of course, this is not really that interesting, since it is really nothing more than the observation that the limit of a functor $F : C rightarrow Set$ can be calculated as the set of sections of the natural projection $int F rightarrow C$, together with definition of the right Kan extension.

On the other hand, it suggests (to me at least) that there is unlikely to be a more "global" description in terms of the Grothendieck construction since the object we are trying to describe is like a "union of a collection of maps," and these to operations tend not to commute (maps from a union is the product of the maps on each component). Probably you knew all this, but maybe somebody will find it useful . . .