Let $F\colon C\to D$ be a functor. Given a functor $\delta\colon D\to Sets$, one can compose with $F$ to get $\delta\circ F\colon C\to Sets$. This process is functorial in the category of $D$-sets, and we denote it $$F^\ast\colon D-set\to C-set.$$ The functor $F^\ast$ has both a left and a right adjoint, denoted $F_!$ and $F_\ast$ respectively.

It turns out that the two left adjoints in this picture, $F^\ast$ and $F_!$ have nice descriptions in terms of the Grothendieck construction. $F^\ast$ is given by pullback: given a functor $\delta\colon C\to Sets$, take its Grothendieck construction $\int\delta\to D$ and form the product with $C\to D$ over $D$; this is $F^\ast\delta$. The "left pushforward" functor $F_!$ is given by a factorization system on $Cat$; given $\gamma\colon C\to Sets$, we have a composition $$\int\gamma\to C\to D$$ which we can factor as an initial functor followed by a discrete op-fibration. The discrete op-fibration is the Grothendieck construction applied to $F_!\gamma$.

Does the "right pushforward" functor $F_\ast$ have any kind of description in terms of Grothendieck constructions? If not, is there a nice reason?