We use the hydrodynamic, cosmological EAGLE simulations to investigate how the hot gas in haloes condenses to form and grow galaxies. We select haloes from the simulations that are actively cooling and study the temperature, distribution and metallicity of their hot, cold and transitioning 'cooling' gas, placing these in the context of semi-analytic models. Our selection criteria lead us to focus on Milky Way-like haloes. We find that the hot-gas density profiles of the haloes form a progressively stronger core over time, the nature of which can be captured by a β profile that has a simple dependence on redshift. In contrast, the hot gas that will cool over a time-step is broadly consistent with a singular isothermal sphere. We find that cooling gas carries a few times the specific angular momentum of the halo and is offset in spin direction from the rest of the hot gas. The gas loses ∼60 per cent of its specific angular momentum during the cooling process, generally remaining greater than that of the halo, and it precesses to become aligned with the cold gas already in the disc. We find tentative evidence that angular-momentum losses are slightly larger when gas cools on to dispersion-supported galaxies. We show that an exponential surface density profile for gas arriving on a disc remains a reasonable approximation, but a cusp containing ∼20 per cent of the mass is always present, and disc scale radii are larger than predicted by a vanilla Fall & Efstathiou model. These scale radii are still closely correlated with the halo spin parameter, for which we suggest an updated prescription for galaxy formation models.