  Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category
with a proper class $\xi$ of $\mathbb{E}$-triangles, and $\mathcal{W}$ an
additive full subcategory of $\mathcal C$. We provide a method for constructing
a proper $\mathcal{W}(\xi)$-resolution (respectively, coproper
$\mathcal{W}(\xi)$-coresolution) of one term in an $\mathbb{E}$-triangle in
$\xi$ from that of the other two terms. By using this way, we establish the
stability of the Gorenstein category $\mathcal{GW}(\xi)$ in extriangulated
categories. These results generalise their work by Huang and Yang-Wang, but the
proof is not too far from their case. Finally, we give some applications about
our main results.
