Natural *-isomorphism

Question

Show that if $X,Y$ are locally compact spaces, then there is a natural *-isomorphism from $C_0(X,C_0(Y))$ onto $C_0(X\times Y)$.

My attempt: I define $\phi:C_0(X,C_0(Y))\to C_0(X\times Y)$ such that $\phi(f)(x,y):=(f(x))(y)$. clearly we can show it's an *-isomorphism. For every $g\in C_0(X\times Y)$, define f(x):=g(x,.) to show that it is onto.

Please check my attempt. Thanks in advance.

Answer 1

Yes, your argument is correct.