nLab
cohomological functor

An additive functor $F$ from a triangulated category $A$ (with translation denoted $X\mapsto X[1]$) to an abelian category $C$ is a cohomological functor if for any distinguished triangle

$$X\stackrel{f}{\to }Y\stackrel{g}{\to }Z\stackrel{h}{\to }X[1],$$X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to X[1],

in $A$, $FX\stackrel{Ff}{\to }FY\stackrel{Fg}{\to }FZ$ is an exact sequence in $C$.