In the book "A course in commutative Banach Algebras," Kaniuth shows that the Gelfand representation of $C_0(X),$ where $X$ is locally compact is the identity map. The argument is as follows:
The map $$x \to M_x=\{f \in C_0(X):f(x)=0\}$$ sets up a one-to-one correspondence between the points of $X$ and the maximal modular ideals of $C_0(X).$ On the other hand, we have a bijection $$\Delta(C_0(X))\to \operatorname{Max}(C_0(X)), \phi\mapsto \ker \phi$$ This gives us a bijection $$X \to \Delta(C_0(X)),x\mapsto \phi_x$$ where $\phi_x$ is defined as $\phi_x(f)=f(x).$ This map is then shown to be a homeomorphism.
Everything is clear up to this point. It is then concluded that
After identifying $X$ with $\Delta(C_0(X))$, the Gelfand representation of $C_0(X)$ is the identity mapping.
This last line is unclear to me. I know identifying $X$ with $\Delta(C_0(X))$ makes the Gelfand representation a map from $C_0(X)$ to $C_0(X).$ But how does it make it the identity map?