I am confused with the underlined equation in the following picture.
[![enter image description here][1]][1]
I know that a *-isomorphism communicates with continuous functional calculus since every continuous functions on the compact subset of $\mathbb{R}$ can be  approximated uniformly by polynomials. 

To prove the underlined equation, we have to prove that $f\to \Phi(f(j_A(a)))$ is the Borel functional calculus on $\Phi(j_A(a))$.
Given a net $(f_\lambda)$ of continuous functions converging to $0$ respect to $\sigma(A^{**},A^*)$ topology , we have $f_\lambda(j_A(a))\xrightarrow{\sigma(A^{**},A^*)}0$, but how to obtain $\Phi(f_\lambda(j_A(a)))\xrightarrow{\sigma(B^{**},B^*)}0$?


Notes: 1.The whole article is [here][2];<br/>
2. My interest is the unital case, you can simply take $M(A)=A$;<br/>
3. $j_A$ in the picture is the canonical injection from $A$ to $A^{**}$.


  [1]: https://i.sstatic.net/5JGb9.png
  [2]: http://www.math.nsysu.edu.tw/~wong/papers/cl-cst-alg-8c.pdf