  W. Paschke's version of Stinespring's theorem associates a Hilbert
$C^*$-module along with a generating vector to every completely positive map.
Building on this, to every quantum dynamical semigroup (QDS) on a $C^*$-algebra
$\mathcal A$ one may associate an inclusion system $E=(E_t)$ of Hilbert
$\mathcal A$-$\mathcal A$-modules with a generating unit $\xi =(\xi_t)$.
Suppose $\mathcal B$ is a von Neumann algebra, consider $M_2(\mathcal B)$, the
von Neumann algebra of $2\times 2$ matrices with entries from $\mathcal B$.
Suppose $(\Phi_t)_{t\ge 0}$ with $\Phi_t=\begin{pmatrix} \phi_t^1& \psi_t
\psi_t^*&\phi_t^2 \end{pmatrix},$ is a QDS on $M_2(B)$ which acts block-wise
and let $(E^i_t)_{t\ge 0}$ be the inclusion system associated to the diagonal
QDS $(\phi^i_t)_{t\ge 0}$ with the generating unit $(\xi_t^i)_{t\ge 0}, i=1,2.$
It is shown that there is a contractive (bilinear) morphism $T=(T_t)_{t\ge0}$
from $(E^2_t)_{t\ge 0}$ to $(E^1_t)_{t\ge 0}$ such that $\psi_t(a)=\langle
\xi^1_t, T_t a\xi^2_t\rangle $ for all $a\in\mathcal B.$ We also prove that any
contractive morphism between inclusion systems of von Neumann $\mathcal
B$-$\mathcal B$-modules can be lifted as a morphism between the product systems
generated by them. We observe that the $E_0$-dilation of a block quantum Markov
semigroup (QMS) on a unital $C^*$-algebra is again a semigroup of block maps.
