  For a quiver $Q$, a $k$-algebra $A$, and a full subcategory $\mathcal X$ of
$A$-mod, the monomorphism category ${\rm Mon}(Q, \mathcal X)$ is introduced.
The main result says that if $T$ is an $A$-module such that there is an exact
sequence $0\rightarrow T_m\rightarrow...\rightarrow T_0\rightarrow
D(A_A)\rightarrow 0$ with each $T_i\in {\rm add} (T)$, then ${\rm Mon}(Q, \
^\perp T) = \ ^\perp (kQ\otimes_k T)$; and if $T$ is cotilting, then
$kQ\otimes_k T$ is a unique cotilting $\m$-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
$(kQ\otimes_kA)$-modules is characterized as ${\rm Mon}(Q, \mathcal{GP}(A))$ if
$A$ is Gorenstein; the contravariantly finiteness of ${\rm Mon}(Q, \mathcal X)$
can be described; and a sufficient and necessary condition for ${\rm Mon}(Q,
A)$ being of finite type is given.
