 Split equalizers associated with a monad

For a monad $(T,\eta ,\mu )$, the map ${\eta }_{TA}:TA\to TTA$ is a split equalizer of ${\eta }_{TTA},T{\eta }_{TA}$ with retraction ${\mu }_A$ and splitting $T{\mu }_A$. The map $T{\eta }_A:TA\to TTA$ is a split equalizer of $TT{\eta }_A,T{\eta }_{TA}$ with retraction ${\mu }_A$ and splitting ${\mu }_{TA}$. Each assertion follows from the two unit laws and naturality of the unit or multiplication. No assumption that $T$ preserves arbitrary equalizers is needed.