A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). Generally, however, even projective resolutions are not guaranteed to be functorial.
Since abelian sheaves form a Grothendieck category, and hence have functorial injective embeddings, I was wondering whether this is enough to actually give a deformation for each functor on the category of (positively graded) chain complexes of abelian sheaves. If so, how does one construct it from functorial embeddings?
Constructing functorial injective resolutions $\mathsf{Ab}(X)\longrightarrow \mathsf{Ch}^{\geq 0}_\bullet (\mathsf{Ab}(X))$ seems easy using splicing and the fact images and kernels are also functors. I don't know how to construct a deformation $\mathsf{Ch}^{\geq 0}_\bullet (\mathsf{Ab}(X))\longrightarrow \mathsf{Ch}^{\geq 0}_\bullet (\mathsf{Ab}(X))$ though, not to mention a deformation for a given functor...
So:
Does each functor between categories of abelian sheaves have a deformation, and how can one construct it from the functorial injective embedding?
Added Later: See Definitions 2.2.1,2.2.4 here.