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We present the Spectroscopic Classification of Astronomical Transients (SCAT) survey, which is dedicated to spectrophotometric observations of transient objects such as supernovae and tidal disruption events. SCAT uses the SuperNova Integral-Field Spectrograph (SNIFS) on the University of Hawai'i 2.2-meter (UH2.2m) telescope. SNIFS was designed specifically for accurate transient spectrophotometry, including absolute flux calibration and host-galaxy removal. We describe the data reduction and calibration pipeline including spectral extraction, telluric correction, atmospheric characterization, nightly photometricity, and spectrophotometric precision. We achieve $\lesssim 5\%$ spectrophotometry across the full optical wavelength range ($3500-9000~Å$) under photometric conditions. The inclusion of photometry from the SNIFS multi-filter mosaic imager allows for decent spectrophotometric calibration ($10-20\%$) even under unfavorable weather/atmospheric conditions. SCAT obtained $\approx 640$ spectra of transients over the first 3 years of operations, including supernovae of all types, active galactic nuclei, cataclysmic variables, and rare transients such as superluminous supernovae and tidal disruption events. These observations will provide the community with benchmark spectrophotometry to constrain the next generation of hydrodynamic and radiative transfer models.