In the 3rd edition of Sakurai Quantum Mechanics, section 8.2 on Dirac's equation, Sakurai writes that the Dirac equation satisfies the continuity equation $$\frac{\partial\rho}{\partial t}+\nabla\cdot\boldsymbol{j}=0,$$ where $$\boldsymbol{j}=\Psi^\dagger\alpha\Psi,$$ and says this is easy to prove by the Schrodinger's equation and its adjoint.
My question is: aren't we doing the Dirac's equation? Why does Sakurai says to use the Schrodinger's equation if the two equation acts on different type of objects? In the previous page Sakurai also confuses me when he says, "which form of the Dirac equation we use etc", he mentions equation 8.1 which is $i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=H|\psi(t)\rangle$ the Schrodinger's equation.
Is there a "Schrodinger's equation" for the Dirac's equation somehow? Thank you for your help, I'm just starting on relativistic QM.
$\textbf{Never mind, I see what he means now. }$