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We introduce the `Resolving supermAssive Black hole Binaries In galacTichydrodynamical Simulations' (RABBITS) series of studies to investigate theorbital evolution of supermassive black holes (SMBHs) during galaxy mergers. Wesimulate both disc and elliptical galaxy mergers using the KETJU code, whichcan simultaneously follow galaxy (hydro-)dynamics and small-scale SMBH dynamicswith post-Newtonian corrections. With our SMBH binary subgrid model, we showhow active galactic nuclei (AGNs) feedback affects galaxy properties and SMBHcoalescence. We find that simulations without AGN feedback exhibit excessivestar formation, resulting in merger remnants that deviate from observedproperties. Kinetic AGN feedback proves more effective than thermal AGNfeedback in expelling gas from the centre and quenching star formation. Thedifferent central galaxy properties, which are a result of distinct AGNfeedback models, lead to varying rates of SMBH orbital decay. In the dynamicalfriction phase, galaxies with higher star formation and higher SMBH massespossess denser centres, become more resistant to tidal stripping, experiencegreater dynamical friction, and consequently form SMBH binaries earlier. As AGNfeedback reduces gas densities in the centres, dynamical friction by starsdominates over gas. In the SMBH hardening phase, compared to ellipticalmergers, disc mergers exhibit higher central densities of newly formed stars,resulting in accelerated SMBH hardening and shorter merger time-scales (i.e.$\lesssim 500$ Myr versus $\gtrsim 1$ Gyr). Our findings highlight theimportance of AGN feedback and its numerical implementation in understandingthe SMBH coalescing process, a key focus for low-frequency gravitational waveobservatories.

RABBITS -- I. The impact of AGN feedback on coalescing supermassive black holes in disc and elliptical galaxy mergers