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 $\Lambda = A\otimes_k kQ/I$, 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 $({\rm smon}(Q,I,\mathcal X), {\rm rep}(Q,I,\mathcal Y))$ is a (complete) hereditary cotorsion pair in $\Lambda$-mod. We also show that $A$ is left weakly Gorenstein if and only if so is $\Lambda$. Provided that $kQ/I$ is non-semisimple, the category $^{\perp}\Lambda$ of semi-Gorenstein-projective $\Lambda$-modules coincides with the category of separated monic representations ${\rm smon}(Q,I,^{\perp}A)$ if and only if $A$ is left weakly Gorenstein.