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Observation 1: I have often read statements like "a contravariant functor $F : C \rightarrow D$ is actually a covariant functor $F: C^{op} \rightarrow D$.

Observation 2 In the context of left-exact functors, some authors define a contravariant left-exact (additive) functor by the property that if $$ A \rightarrow B \rightarrow C \rightarrow 0 $$ is exact in C, then $$ 0 \rightarrow F(C) \rightarrow F(B) \rightarrow F(A) $$ is exact in D.

My issue is now in the reconciliation of observations (1) and (2). In particular, let's consider $C = R-mod$. If we follow the adage from (1) about contravariant functors being covariant functors from the opposite category, then the definition in (2) should be reworded to say:

A contravariant functor $F : C \rightarrow D$ is left-exact if whenever

$$ 0 \rightarrow C \rightarrow B \rightarrow A $$ is exact in $C^{op}$, then $$ 0 \rightarrow F(C) \rightarrow F(B) \rightarrow F(A) $$ is exact in D.

However, as noted in the accepted answer to this question: https://mathoverflow.net/questions/29442/what-is-the-opposite-category-of-the-category-of-modules-or-hopf-algebra-repres, $R-mod^{op}$ is not a category of modules and hence it seems unlikely, even if it made sense to talk about exact sequences and homology in this opposite category (of which I am not sure it does), that a sequence would be exact in this opposite category if and only if it was exact in our original category of R-modules. This seems to make the remark in observation (1) seem to be complete nonsense.

Question Is there a way to formally define left/right-exact contravariant in terms of left/right-exact sequences in the opposite of the domain category? If not, what does it really mean for a "contravariant functor to be a covariant functor from the opposite category"? If yes, then how do we even know how to define homology in an arbitrary opposite category?

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The fact you reference, that the opposite of modules is not a module category, is actually rather esoteric, compared with the fact that it is an abelian category. This is precisely a category in which the notions of homological algebra make sense: the homs are abelian groups, there are biproducts, kernels, and cokernels, every mono is a kernel and every epi a cokernel. With this formalism in place, it is in fact true that $A \to^f B \to^g C$ is exact in any abelian category if and only if $C\to^{g^*}B\to^{f^*}A$ is exact in its opposite, as is shown (unless it's taken for granted) in any book on homological algebra.

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