Special Precovered Categories of Gorenstein Categories

Let $\mathcal{A}$ be an abelian category and $\mathcal{P}(\mathcal{A})$ the subcategory of $\mathcal{A}$ consisting of projective objects. Let $\mathcal{C}$ be a full, additive and self-orthogonal subcategory of $\mathcal{A}$ with $\mathcal{P}(\mathcal{A})$ a generator, and let $\mathcal{G}(\mathcal{C})$ be the Gorenstein subcategory of $\mathcal{A}$. Then the right 1-orthogonal category $\mathcal{G}(\mathcal{C}{)}^{{\mathrm{\perp }}_1}$ of $\mathcal{G}(\mathcal{C})$ is both projectively resolving and injectively coresolving in $\mathcal{A}$. We also get that the subcategory $\text{spc}(\mathcal{G}(\mathcal{C}))$ of $\mathcal{A}$ consisting of objects admitting special $\mathcal{G}(\mathcal{C})$-precovers is closed under extensions and $\mathcal{C}$-stable direct summands (*). Furthermore, if $\mathcal{C}$ is a generator for $\mathcal{G}(\mathcal{C}{)}^{{\perp }_1}$, then we have that $\text{spc}(\mathcal{G}(\mathcal{C}))$ is the minimal subcategory of $\mathcal{A}$ containing $$\mathcal{G}$(\mathscr{C})^{\perp_1}\cup $\mathcal{G}$(\mathscr{C})$withrespecttotheproperty(\ast ),andthat$\spc($\mathcal{G}$(\mathscr{C}))$is$\mathscr{C}$-resolvingin$\mathscr{A}$witha$\mathscr{C}$-propergenerator$\mathscr{C}$.