  For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a
unified form of the indecomposable injective objects in the monomorphism
category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective
objects. As applications, we show that for a given self-injective algebra $A$,
a tilting object in the stable category $\underline{A}$-mod induces a natural
tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$.
We also realize the singularity category of the algebra $kQ\otimes_k A$ as the
stable monomorphism category of the module category of $A$.
