Let $G$ and $H$ be groups written multiplicatively, let $S \subset G$, and suppose $G = \langle S\rangle$, where $\langle S\rangle$ denotes the subgroup of $G$ generated by $S$. If $\varphi,\psi: G \to H$ are group homomorphisms such that $\varphi(s)=\psi(s)$ for every $s \in S$, then $\varphi=\psi$ as functions $G \to H$.