Centralisers of Subsystems of Fusion Systems

When $(S,\mathcal{F},\mathcal{L})$ is a $p$-local finite group and $(T,\mathcal{E},\mathcal{L}_0) \unlhd (S,\mathcal{F},\mathcal{L})$ we define an $S$-centraliser $C_S(\mathcal{L}_0)$ of $\mathcal{L}_0$ in $S$ and observe that this is equivalent to a definition of $C_S(\mathcal{E})$ given by Aschbacher. We also appeal to a result of Gross to give a new proof that there is a unique fusion system $C_{\mathcal{F}}(\mathcal{E})$ on $C_S(\mathcal{E})$.