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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.

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Are sections C2.4-2.5 of Sketches of an Elephant relevant? –  Mike Shulman Mar 26 '13 at 4:37
    
Aha, that is excellent. There is also material in there that answers some other questions I had. I vaguely recalled the notion of a fibration of sites, but I couldn't recall where I had seen it. I should keep the Elephant in front of me, rather than on the top shelf behind me. :-) –  David Roberts Mar 26 '13 at 6:28
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When I get time I can write up a CW answer, or you can put in a short answer, Mike, and I will accept it. –  David Roberts Mar 26 '13 at 6:29
    
On reflection, that section in the Elephant, and the paper of Moerdijk on which it depends, is addressing the opposite question of what I am asking, namely, finding a site in the base topos externalising the internal site. –  David Roberts Apr 5 '13 at 6:59
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