  We study the category O of representations over a shifted Yangian. This
category has a tensor product structure and contains distinguished modules, the
positive prefundamental modules and the negative prefundamental modules.
Motivated by the representation theory of the Borel subalgebra of a quantum
affine algebra and by the relevance of quantum integrable systems in this
context, we prove that tensor products of prefundamental modules with
irreducible modules are either cyclic or co-cyclic. This implies the existence
and uniqueness of morphisms, the R-matrices, for such tensor products. We prove
the R-matrices are polynomial in the spectral parameter, and we establish
functional relations for the R-matrices. As applications, we prove the
Jordan--H\"older property in the category O. We also obtain a proof, uniform
for any finite type, that any irreducible module factorizes through a truncated
shifted Yangian.
