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Let C be small category and A an arbitrary category. Consider the categoryC(A) whose objects are functors from C to A, and whose morphisms are naturaltransformations. Given a functor F : A --> B one obtains an induced functor F_C: C(A) --> C(B) . If A and B are abelian categories, we have that C(A) and C(B)are also abelian, and one has two functors R(F_C) : D(C(A)) --> D(C(B)) and(RF)_ C : C(D(A)) --> C(D (B)). The goals of this paper are: 1) to find arelationship between D (C(A)) and C(D(A)); 2) to relate the functors R(F_C) and(RF)_C. As an application, we prove a version of Mukai's Theorem for quiversheaves.

Derived categories of functors and Fourier--Mukai transform for quiver sheaves