A system with two sets of inputs {u(1), u(2)} and two sets of outputs {y(1), y(2)} is considered. The objective is to construct a dynamic feedback compensator between u(1) and y(1) so that the closed-loop transfer matrix between u(2) and y(2) is diagonal and nonsingular. Internal stability of the closed-loop system is also required. This problem can be regarded as the general version of the standard diagonal decoupling problem where the output to be controlled (decoupled) and the measured output are not distinguished. Along these lines, it is shown that the diagonal causality degree dominance (DCDD), which is the key concept in standard diagonal decoupling problems, also arises in the present context in a modified form. The first modification is due to the constraint of internal stability, which gives rise to a new notion : diagonal stability and causality degree dominance (DSCDD). These two notions are investigated in the first part of the paper. The second modification comes from the cross-channel transfer matrices (between u(1) and y(2) and u(2) and y(1)). This yields another new notion : joint essential stability functions. The solvability conditions of the problem are expressed in terms of the properties of the subrings of the ring of stable and proper rational functions and their quotient rings. These subrings are generated by joint essential stability functions of the cross-channel transfer matrices. The degrees of the joint essential stability functions are called the joint essential stability orders, which are extensions of the well-known essential orders and essential stability orders to the problem considered here.