  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.
