  This work studies the Cauchy problem of a two-component higher order
Camassa-Holm system, which is well-posed in Sobolev spaces
$H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$, $s>\frac{7}{2}$ and its solution
map is continuous. We show that the solution map is H\"{o}lder continuous in
$H^{s}(\mathbb{R})\times H^{s-2}(\mathbb{R})$ equipped with the
$H^{r}(\mathbb{R})\times H^{r-2}(\mathbb{R})$-topology for $1\leq r<s$, and the
H\"{o}lder exponent is expressed in terms of $s$ and $r$.
