Theorem 6.15 in Folland states
Let $p$ and $q$ be conjugate exponents. If $1 < p < \infty$, for each $\phi \in (L^p)^*$ there exists $g \in L^q$ such that $\phi(f) = \int fg$ for all $f \in L^p$, and hence $L^q$ is isometrically isomorphic to $(L^p)^*$. The same conclusion holds for $p = 1$ provided $\mu$ is $\sigma$-finite.
I have gone through the proof of the theorem and it is clear to me that there the map $g \mapsto \phi_g(f) = \int fg$ is indeed a surjection. However, I am having trouble seeing how the second part of the theorem holds, namely that $L^q$ is isometrically isomorphic to $(L^p)^*$. For this we would require some type of uniqueness in that every $\phi \in (L^p)^*$ corresponds to a unique $g \in L^q$, however this was not shown.
How does the injectivity and isometric part of the theorem follow?
If $g_1,g_2 \in L^{q}$ and $\int fg_1=\int fg_2$ for all $f \in L^{p}$ then $\int_E g_1 =\int_E g_2$ for every set $E$ of finite measure. This implies that $g_1=g_2$ almost everywhere.