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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).

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Well, it's true for all $G$ if and only if $l$ is left-exact... – Mike Shulman May 8 '12 at 17:35
True, but in my case, I know for certain that my functor $l$ is not left-exact, since I can exhibit an equalizer it does not preserve. However, I only want it to preserve certain equalizers, and my counterexample is not among them. If it helps, in my case, I know that $l$ does preserve finite products. – David Carchedi May 8 '12 at 20:22

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