A note on split extension classifiers of perfect objects

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