We report on the first calibration of the standard Belle II $B$-flavor tagger using the full data set collected at the $Upsilon(4{rm S})$ resonance in 2019 with the Belle II detector at the SuperKEKB collider, corresponding to 8.7 fb$^{-1}$ of integrated luminosity. The calibration is performed by reconstructing various hadronic charmed $B$-meson decays with flavor-specific final states. We use simulation to optimize our event selection criteria and to train the flavor tagging algorithm. We determine the tagging efficiency and the fraction of wrongly identified tag-side $B$~candidates from a measurement of the time-integrated $B^0-overline{B}^0$ mixing probability. The total effective efficiency is measured to be $varepsilon_{rm eff} = big(33.8 pm 3.6(text{stat}) pm 1.6(text{sys})big)%$, which is in good agreement with the predictions from simulation and comparable with the best one obtained by the Belle experiment. The results show a good understanding of the detector performance and offer a basis for future calibrations.
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