Given sites $(C,J)$ and $(D,K)$, and a functor $f\colon C\to D$ satisfying the covering lifting property:
For every object $c$ of $C$ and $K$-covering sieve $S$ of $f(c)$, there is a $J$-covering sieve $R$ of $c$ such that $f(R)$ refines S,
we get a geometric morphism $F\colon Sh(C) \to Sh(D)$ whose inverse image part is given by precomposition with $f$ followed by sheafification.
I'm interested in when $Sh(C)$ is given by sheaves on an internal site in $D$ (so $F$ is bounded), and when we can find a particularly simple description of this internal site in terms of $f$. Are there any results in this direction on the latter point?
EDIT: Mike points out a relevant section of the Elephant, whereby we should really consider not just a functor $f$ as above, but a fibration of sites, and the geometric morphism it gives rise to. Now instead of extracting an external site of definition for $Sh(C)$ from an internal site of definition in $Sh(D)$ (as $F$ is a bounded geometric morphism), I want to find the internal site of definition for $Sh(C)$ in $Sh(D)$ from the given fibration of sites.