How do people call an additive functor from a triangulated category $C$ to an abelian one that converts distinguished triangles into long exact sequences. Does one usually call a covariant functor of this sort 'homology' and denote it by $H_i$, whereas a contravariant functor is called cohomology and is denoted by $H^i$? For example, if we consider the $i$-th homology (?) of $C$ with respect to a $t$-structure $t$ for it, is it fine to denote it by $H_i^t$?
Upd. Besides, consider a cohomological complex $\dots\to C^0\to C^1\to C^2\to \dots$. Does it have homology ($H_i(C)$) or cohomology ($H^i(C)$)?