In the triangulated category $T$, say we have two distinguished triangles: $$X \xrightarrow{f} Y \rightarrow C_f \rightarrow X[1], $$ $$ X \xrightarrow{g} Y \rightarrow C_g \rightarrow X[1]$$ Of course these data uniquely determines the triangele $$X \xrightarrow{f+g} Y \rightarrow C_{f+g} \rightarrow X[1]$$ The natural question is if one can write down some kind of "reasonable" relationship between $C_{f+g}$ and $C_{f}$ and $C_{g}$? (seems like natural approach $f+g=\triangledown \circ (f\oplus g) \circ \triangle$ does not lead to anything reasonable. Maybe the fact that cone construction is not functorial, does not allow the nice interplay between additive and triangulated structures)
