Let $W$ be a Coxeter group with generators $s_1,s_2,s_3$, where $m(s_1,s_2)=3,m(s_1,s_3)=2$, and $m(s_2,s_3)=\infty$.
I understand that there's a surjective morphism $\varphi\colon W\to PGL(2,\mathbb{Z})$ sending $$ s_1\mapsto\begin{bmatrix} 0 & 1 \\ 1 & 0\end{bmatrix}, s_2\mapsto\begin{bmatrix} -1 & 1 \\ 0 & 1\end{bmatrix}, s_3\mapsto\begin{bmatrix} -1 & 0 \\ 0 & 1\end{bmatrix}. $$
Why is this map injective? I'm reading that if follows from the fact that we have the free product $PSL(2,\mathbb{Z})=\langle\varphi(s_1s_3)\rangle*\langle\varphi(s_1s_2)\rangle$, but this is not clear to me how this is used to prove injectivity.
This is an example in 5.1 of Reflection Groups and Coxeter Groups by James Humphreys.