Separated monic correspondence of cotorsion pairs and semi-Gorenstein-projective modules

Given a finite dimensional algebra $A$ over a field $k$, and a finite acyclic quiver $Q$, let $\mathrm{\Lambda }=A{\otimes }_{k}kQ\mathrm{/}I$, where $kQ$ is the path algebra of $Q$ over $k$ and $I$ is a monomial ideal. We show that $(\mathcal{X},\mathcal{Y})$ is a (complete) hereditary cotorsion pair in $A$-mod if and only if $(\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,\mathcal{X}),\mathrm{r}\mathrm{e}\mathrm{p}(Q,I,\mathcal{Y}))$ is a (complete) hereditary cotorsion pair in $\mathrm{\Lambda }$-mod. We also show that $A$ is left weakly Gorenstein if and only if so is $\mathrm{\Lambda }$. Provided that $kQ\mathrm{/}I$ is non-semisimple, the category ${}^{\perp }\mathrm{\Lambda }$ of semi-Gorenstein-projective $\mathrm{\Lambda }$-modules coincides with the category of separated monic representations $\mathrm{s}\mathrm{m}\mathrm{o}\mathrm{n}(Q,I{,}^{\perp }A)$ if and only if $A$ is left weakly Gorenstein.