This question is closely related to this one: Which limits are preserved by prolongation of presheaves?

Suppose that $r:C \hookrightarrow D$ is a full and faithful functor and $l:D \to C$ is a left-adjoint. Can we say when a given left-exact functor $G:D \to D$ has $l \circ G \circ r$ also left-exact?

I'm ok with assuming here that D is complete and cocomplete (and hence so is C).

all$G$ if and only if $l$ is left-exact... – Mike Shulman May 8 '12 at 17:35