We report on the growth of axial InAs-on-GaAs nanowire heterostructures on silicon by molecular beam epitaxy using 20 nm diameter Au catalysts. First, the growth parameters of the GaAs nanowire segment were optimized to achieve a pure wurtzite crystal structure. Then, we developed a two-step growth procedure to enhance the yield of vertical InAs-on-GaAs nanowires. We achieved 90% of straight InAs-on-GaAs nanowires by further optimizing the growth parameters. We investigated the composition change at the interface by energy dispersive X-ray spectroscopy and the nanowire crystal structure by transmission electron microscopy. The nominal composition of the InAs segment is found to be InxGa1-xAs with x=0.85 and corresponds to 6% of lattice mismatch with GaAs. Strain mapping performed by the geometrical phase analysis of high-resolution images revealed a dislocation-free GaAs/In0.85Ga0.15As interface. In conclusion, we successfully fabricated highly mismatched heterostructures, confirming the prediction that axial GaAs/In0.85Ga0.15As interfaces are pseudomorphic in nanowires below 40 nm diameter.