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The "correct" abstract definition of a derived functor involves the theory of "homotopical categories". An abelian category $\cal A$ does not have enough structure alone to consider homotopical properties. To this end, we usually look at $Ch^+(\cal A)$ or $s\cal A$ of chain complexes in $\cal A$ or simplicial objects in $\cal A$ respectively. We equip these larger categories with the structure of a homotopical category by fixing the lluf subcategories consisting of quasi-isomorphisms or weak homotopy equivalences or quasi-isomorphisms respectively (by the Dold-Kan correspondence, these are essentially identical). In the terminology of Dwyer-Hirschhorn-Kan-Smith, left-exact functors are deformable in the sense that they preserve weak equivalences (that is to say, they are homotopical) on a deformation retract of our original category. So in truth, a derived functor is simply the evaluation of a functor on a suitable approximation of our original object by a homotopically equivalent one on which the functor is well-behaved.

When $\cal A$ has enough structure, we can upgrade the homotopical structure on our homotopical version of $\cal A$ (that is, chain complexes or simplicial objects) to a model structure, which makes it substantially easier to compute derived functors, since the existence of approximations is guaranteed, and further, such approximations can usually be taken to be functorial. For Quillen functors between model categories (these are functors that become homotopical on the deformation retracts consisting of cofibrant or fibrant objects), we can, in the presence of functorial factorization, define the derived functors to be the composition of the functorial approximation and the original functor. These approximations have the property that they descend to the usual Kan extensions at the level of the derived categories (for a proof, see Dwyer-Hirschhorn-Kan-Smith Homotopy Limit Functors on Model Categories and Homotopical Categories).

That is, in your case, we approximate the chain complex $X$ (concentrated in degree zero, so it is constant) by an injective approximation (fibrant for the injective model structure) and evaluate the functor on this new complex. In fact, if injective approximations exist for all chain complexes, we see that when we compose with an appropriate functorial injective approximation, our functor becomes homotopical (it preserves quasi-isomorphisms) and therefore descends uniquely by the universal property of localization to a total derived functor between derived categories of chain complexes.

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The "correct" abstract definition of a derived functor involves the theory of "homotopical categories". An abelian category $\cal A$ does not have enough structure alone to consider homotopical properties. To this end, we usually look at $Ch^+(\cal A)$ or $s\cal A$ of chain complexes in $\cal A$ or simplicial objects in $\cal A$ respectively. We equip these larger categories with the structure of a homotopical category by fixing the lluf subcategories consisting of weak homotopy equivalences or quasi-isomorphisms respectively. In the terminology of Dwyer-Hirschhorn-Kan-Smith, left-exact functors are deformable in the sense that they preserve weak equivalences (that is to say, they are homotopical) on a deformation retract of our original category. So in truth, a derived functor is simply the evaluation of a functor on a suitable approximation of our original object by a homotopically equivalent one on which the functor is well-behaved.

When $\cal A$ has enough structure, we can upgrade the homotopical structure on our homotopical version of $\cal A$ (that is, chain complexes or simplicial objects) to a model structure, which makes it substantially easier to compute derived functors, since the existence of approximations is guaranteed, and further, such approximations can usually be taken to be functorial. For Quillen functors between model categories (these are functors that become homotopical on the deformation retracts consisting of cofibrant or fibrant objects), we can, in the presence of functorial factorization, define the derived functors to be the composition of the functorial approximation and the original functor. These approximations have the property that they descend to the usual Kan extensions at the level of the derived categories (for a proof, see Dwyer-Hirschhorn-Kan-Smith Homotopy Limit Functors on Model Categories and Homotopical Categories).

That is, in your case, we approximate the chain complex $X$ (concentrated in degree zero, so it is constant) by an injective approximation (fibrant for the injective model structure) and evaluate the functor on this new complex. In fact, if injective approximations exist for all chain complexes, we see that when we compose with an appropriate functorial injective approximation, our functor becomes homotopical (it preserves quasi-isomorphisms) and therefore descends uniquely by the universal property of localization to a total derived functor between derived categories of chain complexes.

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The "correct" definition of a derived functor involves the theory of "homotopical categories". An abelian category $\cal A$ does not have enough structure alone to consider homotopical properties. To this end, we usually look at $Ch^+(\cal A)$ or $s\cal A$ of chain complexes in $\cal A$ or simplicial objects in $\cal A$ respectively. We equip these larger categories with the structure of a homotopical category by fixing the lluf subcategories consisting of weak homotopy equivalences or quasi-isomorphisms respectively. In the terminology of Dwyer-Hirschhorn-Kan-Smith, left-exact functors are deformable in the sense that they preserve weak equivalences (that is to say, they are homotopical) on a deformation retract of our original category. So in truth, a derived functor is simply the evaluation of a functor on a suitable approximation of our original object by a homotopically equivalent one on which the functor is well-behaved.

When $\cal A$ has enough structure, we can upgrade the homotopical structure on our homotopical version of $\cal A$ (that is, chain complexes or simplicial objects) to a model structure, which makes it substantially easier to compute derived functors, since the existence of approximations is guaranteed, and further, such approximations can usually be taken to be functorial. For Quillen functors between model categories (these are functors that become homotopical on the deformation retracts consisting of cofibrant or fibrant objects), we can, in the presence of functorial factorization, define the derived functors to be the composition of the functorial approximation and the original functor. These approximations have the property that they descend to the usual Kan extensions at the level of the derived categories (for a proof, see Dwyer-Hirschhorn-Kan-Smith Homotopy Limit Functors on Model Categories and Homotopical Categories).
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