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This question is mainly posed out of sheer curiosity rather than any practical applications.

An additive functor $F:C\to D$ is called half-exact if for every short exact sequence $$ 0\to L\to M\to N\to 0 $$ There is an exact sequence $$ FL\to FM\to FN $$ When $F$ is in fact left exact (resp. right exact), one can then define the right derived functors (resp. left derived functors) $R^i F$ (resp. $L_i F$). These create long exact sequences which "fill in" the missing exactness in a "natural" way (i.e., in such a way that morphisms of short exact sequences induce morphisms of long exact sequences).

I am interested in the generalization to the case where $F$ is neither left exact nor right exact. Namely: do there exist derived functors $R^i F$ and $L_i F$ such that i) $R^i F = L_{-i} F$, and ii) for every short exact sequence of objects in $C$ there is an induced long exact sequence $$ ...\to L_1 FN\to L\to M\to N\to R_1 FL\to ... $$ which is natural in the sense mentioned above? And if so, how would one go about defining these functors?

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