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The $\alpha$-$z$ R\'enyi relative entropies are a two-parameter family ofR\'enyi relative entropies that are quantum generalizations of the classical$\alpha$-R\'enyi relative entropies. In \cite{zhang20CFL} we decided the fullrange of $(\alpha,z)$ for which the Data Processing Inequality (DPI) is valid.In this paper we give algebraic conditions for the equality in DPI. For thefull range of parameters $(\alpha,z)$, we give necessary conditions andsufficient conditions. For most parameters we give equivalent conditions. Thisgeneralizes and strengthens the results of Leditzky, Rouz{\'e} and Datta in\cite{LRD17DPI}.

Equality conditions of Data Processing Inequality for $\alpha$-$z$ R\'enyi relative entropies