We study the formation of local magnetic moments in the strongly correlated Hubbard model within dynamical mean-field theory and associate peculiarities of the temperature dependence of local charge ${\chi }_c$ and spin ${\chi }_s$ susceptibilities with different stages of local moment formation. The local maximum of the temperature dependence of the charge susceptibility ${\chi }_c$ is associated with the beginning of local magnetic moment formation, while the minimum of the susceptibility ${\chi }_c$ and double occupation, as well as the low-temperature boundary of the plateau of the effective local magnetic moment ${\mu }_{\mathrm{eff}}^2=T{\chi }_s$ temperature dependence are connected with the full formation of local moments. We also obtain the interaction dependence of the Kondo temperature $T_K$, which is compared to the fingerprint criterion of Chalupa et al. [Phys. Rev. Lett. 126, 056403 (2021)]. Near the Mott transition the two criteria coincide, while further away from the Mott transition the fingerprint criterion somewhat overestimates the Kondo temperature. The relation of the observed features to the behavior of eigenvectors/eigenvalues of fermionic frequency-resolved charge susceptibility and divergences of irreducible vertices is discussed.

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• Kondo effect
• Many-body localization
• Metal-insulator transition
• Dynamical mean field theory