this is my first thread on mathoverflow, and apologies if this is a trivial question.
In Dwyer and Spalinski's Homotopy theories and model categories, they gave the definition of derived functors, and based on their definition, we require the source C of a functor F : C $\to$ D to be a model category, and if a derived functor (no matter left or right) exists, denoted as DF, then
DF : Ho(C) $\to$ D
The notes also gives a sufficient condition of the existence of left derived functor, but because of duality, we can rephrase the same for right derived functor: if the functor F is as above, and F(f) is an isomorphism whenever f is a weak equivalence between fibrant objects in C, then the right derived functor (RF, t) of F exists.
I think I accept the concepts above fairly well, but then in Section 3.1 of Hartshorne's Algebraic Geometry, he introduced the concept of derived functors of covariant left exact functors F : A $\to$ B, where A and B are abelian categories, with A having enough injectives. Though Hartshorne didn't mention the condition of existence, he wrote if F: A $\to$ B is a covariant left exact functor between abelian categories, and A has enough injectives, then we construct the right derived functor...
So, does the concepts in the two scenarios above correspond? For example (this might be very wrong, just my guess), does "an abelian category with enough injectives" automatically have a model structure?
If the answer is "yes" and we have a model structure compatible with the cohomology objects, does "a covariant left exact functor" correspond to "a functor sending fibrant objects to isomorphisms", and do other properties of derived functors, such as the universal property, also hold?
If the answer is "no", then does it mean we can construct derived functors without model structure and the associated homotopy category? If so, how? Also, are there any relationships between model categories and abelian categories?
Thank you very much for reading my long and tedious thread. I appreciate any insights.