  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.
