In this work we develop a new method to turn a state-of-the-art hydrodynamical cosmological simulation of galaxy formation (HYD) into a simple semi-analytic model (SAM). This is achieved by summarizing the efficiencies of accretion, cooling, star formation and feedback given by the HYD, as functions of the halo mass and redshift. The SAM then uses these functions to evolve galaxies within merger trees that are extracted from the same HYD. Surprisingly, by turning the HYD into a SAM, we conserve the mass of individual galaxies, with deviations at the level of 0.1 dex, on an object-by-object basis, with no significant systematics. This is true for all redshifts, and for the mass of stars and gas components, although the agreement reaches 0.2 dex for satellite galaxies at low redshift. We show that the same level of accuracy is obtained even in case the SAM uses only one phase of gas within each galaxy. Moreover, we demonstrate that the formation history of one massive galaxy provides sufficient information for the SAM to reproduce the population of galaxies within the entire cosmological box. The reasons for the small scatter between the HYD and SAM galaxies are as follows. (i) The efficiencies are matched as functions of the halo mass and redshift, meaning that the evolution within merger trees agrees on average. (ii) For a given galaxy, efficiencies fluctuate around the mean value on time-scales of 0.2-2 Gyr. (iii) The various mass components of galaxies are obtained by integrating the efficiencies over time, averaging out these fluctuations. We compare the efficiencies found here to standard SAM recipes and find that they often deviate significantly. For example, here the HYD shows smooth accretion that is less effective for low-mass haloes, and is always composed of hot or dilute gas; cooling is less effective at high redshift, and star formation changes only mildly with cosmic time. The method developed here can be applied in general to any HYD, and can thus serve as a common language for both HYDs and SAMs.