Hovey triples arising from two cotorsion pairs of extriangulated categories

Assume that (C, E, s) is an extriangulated category satisfying Condition (WIC). Let (Q, (R) over tilde) and ((Q) over tilde, R) be two hereditary cotorsion pairs with (R) over tilde subset of R, (Q) over tilde subset of Q and (Q) over tilde boolean AND R = Q boolean AND (R) over tilde. Then there exists a unique thick class W for which (Q, W, 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.