  For an abelian category $\mathcal{A}$ we investigate when the stable
categories $\underline{\mathrm{GPro}}\mathrm{j}(\mathcal{A})$ and
$\underline{\mathrm{GIn}}\mathrm{j}(\mathcal{A})$ are triangulated equivalent.
To this end, we realize these stable categories as homotopy categories of
certain (non-trivial) model categories and give conditions on $\mathcal{A}$
that ensure the existence of a Quillen equivalence between the model categories
in question. We also study when such a Quillen equivalence transfers from
$\mathcal{A}$ to categories naturally associated to $\mathcal{A}$, such as
$\mathrm{Ch}(\mathcal{A})$, the category of chain complexes in $\mathcal{A}$,
or $\mathrm{Rep}(Q,\mathcal{A})$, the category of $\mathcal{A}$-valued
representations of a quiver $Q$.
