Doubt about Proposition 2.39 in Dana Williams' crossed product book You can see the proposition in a google books preview here. First and foremost, my question is:

Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence between 
  
  
*
  
*The set of nondegenerate covariant homomorphisms  $(\pi,u) : (A,G,\alpha) \to \mathcal{L}(X)$.
  
*The set of nondegenerate homomorphisms $L : A \rtimes_\alpha G \to \mathcal{L}(X)$.
  
  
  by sending each $(\pi,u)$ to $L = \pi \rtimes u$ and sending each $L$ to the $(\pi,u)$ defined by $\pi(a) = \overline{L}(i_A(a)), u_s = \overline{L}(i_G(s))$ for all $a \in A, s \in G$. 

I don't really have a mathematical reason to doubt this reading is correct. My reasons are slightly meta, so I hope I am making myself understood. The immediately following Proposition 2.40 would seem, in part, to be a corollary, where the Hilbert $B$-module $X$ is taken to be a Hilbert space.  
Now, in Proposition 2.40 Williams states explicitly that a bijective correspondence is being set up, whereas in Proposition 2.39 this is only implicit. That in and of itself would not be enough to raise my eyebrows, but, have a look at the 1st paragraph of the proof of Proposition 2.40. 

"Proposition 2.39 on the facing page shows that the map $(\pi,U) \mapsto \pi \rtimes U$ is a surjection. It's one-to-one in view of Equations (2.21) and (2.27)." 

I don't understand the need for the reference to equations (2.21) and (2.27). Does Proposition 2.40 not already show we have a bijection?
 A: Let $ (A,G,\alpha) $ be a $ C^{*} $-dynamical system. Let $ B $ be a $ C^{*} $-algebra, and fix a Hilbert $ B $-module $ \mathsf{X} $ that will serve as a representation space. Proposition 2.39 then says that the association
\begin{align}
\{ \text{Covariant representations of $ (A,G,\alpha) $ on $ \mathsf{X} $} \} & \to
\{ \text{$ * $-representations of $ A \rtimes_{\alpha} G $ on $ \mathsf{X} $} \} \\
(\pi,U) & \mapsto \pi \rtimes_{\alpha} U
\end{align}
is surjective, where covariant representations and $ * $-representations are implicitly assumed to be non-degenerate. More precisely, the proposition says that if
$$
L: A \rtimes_{\alpha} G \to \mathcal{L}(\mathsf{X})
$$
is a $ * $-representation of $ A \rtimes_{\alpha} G $ on $ \mathsf{X} $, then $ L = \pi \rtimes_{\alpha} U $, where $ \pi = \bar{L} \circ i_{A} $ and $ U = \bar{L} \circ i_{G} $.
(Note: $ \bar{L} $ is the canonical extension of $ L $ to $ M(A \rtimes_{\alpha} G) $, and $ i_{A} $ and $ i_{G} $ are, respectively, the canonical inclusions of $ A $ and $ G $ into $ M(A \rtimes_{\alpha} G) $.)
Notice that the proposition does not imply that the association described above is injective. For all we know, there might very well be another covariant representation $ (\rho,V) $ of $ (A,G,\alpha) $ on $ \mathsf{X} $ such that $ L = \rho \rtimes_{\alpha} V $. Fortunately for us, Equations 2.21 and 2.27 rule out this possibility.
To see how these two equations work to establish injectivity, suppose that $ (\pi,U) $ is a covariant representation of $ (A,G,\alpha) $ on $ \mathsf{X} $. Letting $ L = \pi \rtimes_{\alpha} U $, Equations 2.21 and 2.27 respectively yield
\begin{align}
  \forall a \in A, ~ \forall \phi \in {C_{c}}(G,A): \quad
& L([{i_{A}}(a)](\phi)) = \pi(a) \circ L(\phi), \\
  \forall g \in G, ~ \forall \phi \in {C_{c}}(G,A): \quad
& L([{i_{G}}(g)](\phi)) = U_{g} \circ L(\phi),
\end{align}
where $ {C_{c}}(G,A) $ is being viewed as a dense $ * $-subalgebra of $ A \rtimes_{\alpha} G $. As $ L $ is a non-degenerate $ * $-representation, the set
$$
\{ [L(\phi)](x) \mid \phi \in {C_{c}}(G,A) ~ \text{and} ~ x \in \mathsf{X} \}
$$
is dense in $ \mathsf{X} $. It follows that $ \pi(a) $ and $ U_{g} $ are uniquely determined for each $ a \in A $ and each $ g \in G $. Therefore, $ (\pi,U) $ is the unique covariant representation of $ (A,G,\alpha) $ on $ \mathsf{X} $ whose integrated form is $ L $.
