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5. The torsion functors Tor are the derived functors of the tensor product.
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Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product.
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1955 Alexander Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
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The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.
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Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors LnT are defined.
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Now the obvious uniqueness of f' suffices to obtain the required adjunction—the morphism-part of the functor F can be derived from general considerations (see adjoint functors).
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From the category theory point of view, a state monad is derived from the adjunction between the product functor and the exponential functor, which exists in any cartesian closed category by definition.
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