A question in the proof that the map $\gamma: \phi \to \phi^*$ from $Reg(V,W)$ to $Hom_{k-alg }( \Gamma(W), \Gamma(V))$ is bijective

I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry.

$k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V,W$ is affine algebraic set. $\theta : \Gamma(W) \to \Gamma(V)$ be a homomorphism of $k-$algebras and $\phi: V\to k^m$ is the map whose co-ordinates are $\phi_i$. If $\phi:V\to W$be a morphism such that $V\subset k^n $ and $W\subset k^m$ written in the form $\phi= (\phi_1,...,\phi_m)$ , where $ \phi_i \in \Gamma(V)$. Set $\phi_i =\theta(\eta_i)\in \Gamma(V)$ Here $ \eta_i$ is the $i$ th coordinate function of $W$, which is the image of the variable $y_i$ in $\Gamma(W)$

Page $21$-Page $22$ of Daniel Perrin's Algebraic Geometry.

Defintion: Let $\phi:V\to W$ be a morphism. For any $f\in \Gamma(W)$, set $\phi^*(f)= f\circ \phi$. Then $\phi^*$ is a morphism of $k-$algebras from $\Gamma(W) \to \Gamma(V)$. $\phi^*$ is denoted by $\Gamma$

Prop$.6.7: $ The functor $\Gamma$ is fully faithful ie the map $\gamma: \phi \to \phi^*$ from $Reg(V,W)$ to $Hom_{k-alg }( \Gamma(W), \Gamma(V))$ is bijective

Proof: I understand how $\Gamma$ is injective.

I understand how the author shows the image of $\phi$ is contained in $W$.

But I donot understand how showing that the image of $\phi$ is contained in $W$ implies $\theta= \phi^*$ and which shows the surjectivity of $\gamma$?

Can you please help me with understanding?