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Let $\mathcal{A}$ be an abelian category with enough injectives and projectives. Let $T: \mathcal{A} \to$ Ab be an additive functor into the category of abelian groups. You can assume that $T$ is left or right exact, if you wish. In Joseph Rotman's text on homological algebra, and in the other sources that I have searched, there are exactly three ways to take the derived functors of $T$. If $T$ is covariant, then you can obtain its left or right derived functors by taking either projective or injective resolutions of your objects. If $T$ is contravariant, you can play the same game with projective resolutions to obtain its right derived functors.

What no one seems to consider is taking the left derived functor of an additive contravariant functor via injective resolutions. There should be no issues with the definition of these functors, although I confess I have not checked that this gives you a $\delta$-functor. Do this construction fail to give you a $\delta$-functor, or is there some other reason for why no one seems to care about this fourth method for taking derived functors?

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You can take the left derived functor of an additive contravariant functor, since it can also be viewed as an additive covariant functor $\mathcal A^{\mathrm{op}}\to\mathrm{Ab}$, where $\mathcal A^{\mathrm{op}}$ is the opposite category of $\mathcal A$. However, we usually only apply left derived functors when $T$ is right exact, so it's possible that case was left out because right exact contravariant functors are rare compared to the other three kinds.

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  • $\begingroup$ I agree with your answer, but I think it is worth mentioning that one does sometimes care about left-derived functors of functors that are not right-exact; for example, if $(A,\mathfrak{m})$ is a complete local ring, then the left-derived functors of $\mathfrak{m}$-adic completion show up occasionally. $\endgroup$ – ne- Jul 17 '18 at 17:42

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