\displaystyle\begin{aligned}\left[g\cdot(h\cdot f)\right](v)&=g\left[h\cdot f\right](g^{-1}v)\\&=g(hf(h^{-1}(g^{-1}v)))\\&=(gh)f((h^{-1}g^{-1})v)\\&=(gh)f((gh)^{-1}v)\\&=\left[(gh)\cdot f\right](v)\end{aligned}