Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

Question

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone correspondence.

Let $\gamma : S_{\sigma} \to \mathbb{C}$ be a semigroup homomorphism giving a point of $U_{\sigma}$. Prove that $S' = \{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\} = \Gamma \cap M$ for some face $\Gamma \preceq \sigma^V$.

I think it is correct to say $\sigma^{\bot} \cap M \subset S'$ indeed if $m \in \sigma^{\bot} \cap M$ also $-m \in\sigma^{\bot} \cap M$ then $\gamma(m) = 0$ would imply $\gamma(-m) = (\gamma(m))^{-1} = 0^{-1}$ but zero is not invertible. So $m \in \sigma^{\bot} \cap M$ implies $\gamma(m) \neq 0$ and hence the inclusion.

But then I don't know how to proceed to conclude the thesis.