Normalizers and centralizers of $p$-subgroups in normal subsystems of fusion systems

Suppose $mathcal{E}$ is a normal subsystem of a saturated fusion system $mathcal{F}$ over $S$. If $Xleq S$ is fully $mathcal{F}$-normalized, then Aschbacher defined a normal subsystem $N_{mathcal{E}}(X)$ of $N_{mathcal{F}}(X)$. In this short note we revisit and generalize this result using the theory of localities. Our more general approach leads in particular to a normal subsystem $C_{mathcal{E}}(X)$ of $C_{mathcal{F}}(X)$ for every $Xleq S$ which is fully $mathcal{F}$-centralized.