...¹

sup of objects is also written as sum

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...²

this is equivalent to saying injective resolution exists, which by definition, is exact sequence $0\rightarrow A\rightarrow I_0\rightarrow I_1 \rightarrow \ldots$ where each $I_i$ is injective object.

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...³

this holds if category has enough injectives

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...⁴

We are here concerned with covariant functors only and what we call derived functor is right derived functor, as defined in literature. Since it is only matter of reversing arrows, we wont define left derived functors of contravariant functor here.

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...⁵

since in degree 0, $S^0F(A) = F(A)$, we have $F(A) \rightarrow Ker(F(I^0)\rightarrow F(I^1)) = R^0F(A)$

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