I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.

The question is: given a tranformation $F\to G$ of two functors of this sort, is it possible to extend it to a long exact sequence $\dots \to H\circ[1]\to F\to G\to H\to F\circ [-1]\to G\circ[-1]\to \dots$? Here $H$ should be cohomological also, and a sequence is called exact if it is exact when applied to any object of $C$.

Note that in the case where $C$ is a "countable" subcategory of compact objects in a certain triangulated $C'$ that is closed with respect to coproducts one has the Neeman's Brown representability for homology property (for $C\subset C'$; see the corresponding Neeman's papers); thus $F$ and $G$ are representable and any transformation between them can be lifted to a $C'$-morphism. Hence one can just take $H$ represented by a cone of (any possible) morphism of this sort.

Yet it does not seem that constructing cones in the "uncountable" case is hopeless. Did anybody study related problems before? Any hints would be very welcome!

Upd. Actually, I am mostly interested in the following particular case of my question: does $H$ exist if $F,G$ are ind-representable functors; can $H$ be chosen to be ind-representable also in this case?