For two bimodules NBA and MAB with M⊗AN=0=N⊗BM, the monomorphism category M(A,M,N,B) and its dual, the epimorphism E(A,M,N,B), are introduced and studied. By definition, M(A,M,N,B) is the subcategory of Δ-mod consisting of (X,Y,f,g) such that f:M⊗AX→Y is a monic B-map and g:N⊗BY→X is a monic A-map, where Δ=(ANMB) is a Morita ring. This monomorphism category is a resolving subcategory of Δ-mod if and only if MA and NB are projective modules; and if this is the case and if in addition HomB(N,B)A and HomA(M,A)B are projective modules, then it has enough relative injective objects, and it is a Frobenius category if and only if A and B are selfinjective. Under these conditions we prove that there is a unique cotilting left Δ-module T, up to the multiplicity of the indecomposable direct summands, such that M(A,M,N,B)=T⊥. When M and N are exchangeable bimodules, Ringel-Schmidmeier-Simson equivalence between M(A,M,N,B) and E(A,M,N,B) are given.