Hovey triples arising from two cotorsion pairs of extriangulated categories

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