= Solution
Use $g^{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ and restore $\hbar$ temporarily. The free <Dirac equation> is $(i\hbar\gamma^\mu\partial_\mu-m)\psi=0$, with $c=1$. Its Hermitian conjugate is $-i\hbar(\partial_\mu\psi^\dagger)(\gamma^\mu)^\dagger-m\psi^\dagger=0$. Multiplying on the right by $\gamma^0$ and using $(\gamma^\mu)^\dagger\gamma^0=\gamma^0\gamma^\mu$ gives the adjoint equation
$$
i\hbar(\partial_\mu\bar\psi)\gamma^\mu+m\bar\psi=0,\qquad \bar\psi=\psi^\dagger\gamma^0.
$$
The <Dirac adjoint> and original equations give $(\partial_\mu\bar\psi)\gamma^\mu\psi=im\bar\psi\psi/\hbar$ and $\bar\psi\gamma^\mu\partial_\mu\psi=-im\bar\psi\psi/\hbar$. Adding gives the <continuity equation>
$$
\partial_\mu j^\mu=0,\qquad j^\mu=\bar\psi\gamma^\mu\psi,\qquad j^0=\psi^\dagger\psi\geq0,\qquad \mathbf j=\psi^\dagger\boldsymbol\alpha\psi.
$$
The <Dirac current> transforms as a <four-vector>; the positive density is the time component, not a <Lorentz scalar>. With ordinary units the number current is $c\bar\psi\gamma^\mu\psi$ and the <electron> charge current is $-e$ times it.
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