Map having a cokernel in a triangulated category

Question

Let $\mathcal{T}$ be a triangulated category and $f:X\rightarrow Y$ be a map in $\mathcal{T}$. I think that I can prove that if $f$ has a cokernel, then $f$ can be decomposed as a map $X\cong A\oplus B\rightarrow B\oplus coker(f)\cong Y$, in which the middle map is given by $\matrix{0\ 0\\1\ 0}$. Is this actually true?