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 $X\le S$ 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 $X\le S$ which is fully $\mathcal{F}$-centralized.