  We show that for a pointed protomodular category $\mathbb{C}$ satisfying a
certain condition on those Huq commutators which exist, if $X$ is a perfect
object in $\mathbb{C}$ such that the split extension classifier $[X]$ exists,
then the centralizer of the \emph{conjugation} morphism $c_X : X\to [X]$ is
trivial and hence $[X]$ has trivial center.
