# Double adjoint map in C*-algebra

There is a quotation below:

Assume $A$ is nonunital C*-algebra and $B$ is unital C*-algebra and $\phi: A\rightarrow B$ is a contractive completely positive map.

Consider the double adjoint map $\phi^{**}:A^{**}\rightarrow B^{**}$. Identifying double duals with enveloping von Neumann algebras, one checks that maps positive operators to positive operators. Since $M_{n}(C^{**})\cong(M_{n}(C))^{**}$ for any C*-algebra $C$, it follows that $\phi^{**}$ is also completely positive.

My question is: How to verify that $\phi^{**}$ is completely positive?

Because now $(\phi^{**})^{(n)}:M_n(A^{**})\to M_n(B^{**})$ can be seen as $(\phi^{(n)})^{**}:M_n(A)^{**}\to M_n(B)^{**}$. As $\phi^{(n)}$ is positive (because $\phi$ is completely positive), $(\phi^{(n)})^{**}$ is positive by the fact that the double dual map of a positive map is positive.