Past exam of the short options course of the University of Oxford / 2023 / 2 / c / ii / Solution

The identity $T=S^{-1}(ST)$ shows that $S,ST$ generate the same group. Direct multiplication givesConsequently their images $\bar{S},\overline{ST}$ in the quotient group $PSL(2,\mathbb{Z})$ satisfy ${\bar{S}}^2=1$ and ${\overline{ST}}^3=1$. The universal property of a group presentation gives a group homomorphism taking $u\mapsto \bar{S}$, $v\mapsto \overline{ST}$ from the stated presentation. It is surjective because these images generate the target. Injectivity is not needed here.