How to prove that $\operatorname{Hom}_{Var} (X,Y) \cong \operatorname{Reg}(X,Y)$

I am self studying algebraic geometry from the textbook of Daniel Perrin (Algebraic Geometry: An Introduction).

On page 44 is the Proposition 3.5 which I am unable to prove and need help with.

Proposition 3.5. Let $(X,O_X)$ and $(Y,O_Y)$ be two affine algebraic sets equipped with the affine variety structures given by structures sheaves $O_X$ and $O_Y$. Then there is natural bijection $\operatorname{Hom}_{Var}(X,Y) \cong \operatorname{Reg}(X,Y)$

Here, $\operatorname{Hom}_{Var}(X,Y)$ is the set of affine variety morphisms from $X$ to $Y$, and $\operatorname{Reg}(X,Y)$ is the set of regular functions from $X$ to $Y$. We denote by $n_i$ the $i $th coordinate function on $W$, which is the image of the variable $Y_i$ in $\Gamma(W)$ then $\phi^{*}(n_i)=\phi_i$

Proof given in the textbook: We establish the existence of bijection as follows: If $\phi: X\to Y$ is a variety morphism, then on considering the co-ordinate functions $n_i$ on $Y$ it is clear that $\phi= (n_1\phi,...,n_m\phi)$ is regular. Conversely, if $\phi$ is regular , $D(g)$ is the standard open set of $Y$ and $f=h/g^r \in \Gamma(D(g),O_Y)$, then $f\phi=$$\phi^*(h)$/${\phi^*(g)}^r \in \Gamma(D(\phi^*(g),O_X)$, which shows that $\phi$ is also a morphism of varieties.

The given proof omits most of the details (as a lot of proofs in the book) and I really need help with proving it.

I shall be extremely grateful!