Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa.
Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. For an automophism $\alpha\in Aut(A)$, the following statements are true:
(1) $A\rtimes_{\alpha} \mathbb{Z}=A\rtimes_{\alpha, r}\mathbb{Z}$.
(2) ....
Proof. Proof of (1): It suffices to show that there exsit c.c.p. (contractive completely positive) maps $$\Psi_{n}:A\rtimes_{\alpha,r}\mathbb{Z}\rightarrow A\rtimes_{\alpha}\mathbb{Z}$$ such that $||x-\Psi_{n}\circ\pi(x)||_{u}\rightarrow0$ for all $x\in C_{c}(\mathbb{Z}, A)\subset A\rtimes_{\alpha} \mathbb{Z}$, where $$\pi:A\rtimes_{\alpha}\mathbb{Z}\rightarrow A\rtimes_{\alpha,r}\mathbb{Z}$$ is the canonical quotient map(coming from universality).
.....
My question is, in order to prove the isomorphism , why does the author find such a c.c.p map $\Psi_{n}$.
In the Theorem, the $A\rtimes_{\alpha}\Gamma$ denotes the full crossed product of a C*-dynamical system $(A, \alpha, \Gamma)$, which is the completion of $C_{c}(\Gamma, A)$ with respect to the norm $$||x||_{u}=sup||\pi(x)||,$$ where the supremum is over all (cyclic) *-homomorphisms $\pi:C_{c}(\Gamma, A)\rightarrow B(H)$. While the $A\rtimes_{\alpha,r}\Gamma$ denotes the reduced crossed product of a C*-dynamical system $(A, \alpha, \Gamma)$, which is the norm closure of the image of a regular representation $C_{c}(\Gamma, A)\rightarrow B(H\otimes l^{2}(\Gamma))$. In addition, the $C_{c}(\Gamma, A)$ denotes the linear space of finitely supported functions on $\Gamma$ with values in $A$.
I think it is simply this fact: take $x\in\ker\pi$. From the relation in the first paragraph of the proof, $$ x=\lim_n\Psi_n(\pi(x))=\lim_n\Psi(0)=0. $$ So $\pi$ is an isometric epimorphism, and so an isomorphism.