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Let 
$\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the 
${{C}_{0}}$ coarse structures. We show that if 
$X$ and 
$Y$ are in $\mathbf{TB}$, then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories 
$\mathbf{TB}\,\to \,\mathbf{K}$, where 
$\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification 
$\widetilde{X}$ is determined only by the topology of the remainder 
$\widetilde{X}\,\backslash \,X$.
