A Hovey triple arising from two cotorsion pairs

Assume that $(\mathcal{C}, \mathbb{E}, \mathfrak{s})$ is an extriangulated category satisfying Condition (WIC). Let $(\cal Q, \widetilde{\cal R})$ and $(\widetilde{\cal Q}, \cal R)$ be two hereditary cotorsion pairs satisfying conditions $\widetilde{\cal R} \subseteq \cal R$, $\widetilde{\cal Q}\subseteq \cal Q$ and $\widetilde{\cal Q}\cap \cal R = \cal Q \cap \widetilde{\cal R}$. Then there exists a unique thick class $\cal W$ for which $(\cal Q,\cal W,\cal R)$ is a Hovey triple. As an application, this result generalizes the work by Gillespie in an exact case. Moreover, it highlights new phenomena when it applied to triangulated categories.