Monomorphism operator and perpendicular operator

For a quiver $Q$, a $k$-algebra $A$, and a full subcategory $\mathcal{X}$ of $A$-mod, the monomorphism category $\mathrm{M}\mathrm{o}\mathrm{n}(Q,\mathcal{X})$ is introduced. The main result says that if $T$ is an $A$-module such that there is an exact sequence $0\rightarrow T_m\rightarrow…\rightarrow T_0\rightarrow D(A_A)\rightarrow 0$witheach$T_i\in {\rm add} (T)$,then${\rm Mon}(Q,
^\perp T) = \ ^\perp (kQ\otimes_k T)$;andif$T$iscotilting,then$kQ\otimes_k T$isauniquecotilting$\m$-module, up to multiplicities of indecomposable direct summands, such that ${\rm Mon}(Q, \ ^\perp T)= \ ^\perp (kQ \otimes_k T)$. As applications, the category of the Gorenstein-projective $(kQ{\otimes }_{k}A)$-modules is characterized as $\mathrm{M}\mathrm{o}\mathrm{n}(Q,\mathcal{G}\mathcal{P}(A))$ if $A$ is Gorenstein; the contravariantly finiteness of $\mathrm{M}\mathrm{o}\mathrm{n}(Q,\mathcal{X})$ can be described; and a sufficient and necessary condition for ${\rm Mon}(Q, A)$ being of finite type is given.