We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $\UT(n,F)$ over a field $F$, where $n\in\N\cup\{\infty\}$. PC-maps were recently described up to almost identity PC-maps by D.Wang, S.Ou, and W.Zhang for finite $n$ and by R.Slowik for $n=\infty$. An almost identity map is a map, preserving elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if $n=\infty$, then an almost identity map is identity. Together with the result of R.Slowik this shows that any PC-map $\UT(\infty, F)\to\UT(\infty, F)$ is an autmorphism.