Before looking at the book's proof, I tried doing the proof myself. After comparing the two I found mine to be a different than the book's. Please check to see if my reasoning is correct.
We have $f:A \rightarrow B$ and $g:B \rightarrow C$. Since $f$ is injective, we can say that $f(x)$ is different for all distinct $x∈A$. Let the set S be the set of all elements in B that can be represented as $f(x)$ . We know that all distinct $y∈B$ have distinct $g(y) ∈ C$ thus all distinct $z∈S$ have a distinct $g(z)∈C$. Thus we can say that for all distinct $x∈A$ have a distinct $g(f(x))$
Please verify if my reasoning is correct and the proof is rigorous enough.