  We study isometric representations of product systems of correspondences over
the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional,
fully coisometric representations. We show the existence of a unique minimal
cyclic coinvariant subspace for all such representations. The compression of
the representation to this subspace is shown to be complete unitary invariant.
For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closed
algebra generated by these representations is shown to contain the projection
onto the minimal cyclic coinvariant subspace. This class includes free
semigroup algebras. This result extends to a class of higher-rank graph
algebras which includes higher-rank graphs with a single vertex.
