Given a (Dirac), spinor in the Weyl basis, $\psi = \begin{pmatrix} \psi_{L}\\ \psi_{R} \end{pmatrix} $ , where $\psi_{L}$ and $\psi_{R}$ are Weyl spinors we define the adjoint of the Dirac spinor as;

$\bar{\psi}=\psi^{\dagger}\gamma^{0}=(\psi_{R}^{\dagger},\psi_{L}^{\dagger})$

I understand this, but recently I've ran into expressions like $\bar{\psi}_{L}$, and $\bar{\psi}_{R}$. I can't seem to find a definition for the adjoint of a Weyl spinor so this is confusing me. Is it as simple as; $$\bar{\psi}_{L}=P_{L}\bar{\psi}=\psi^{\dagger}\gamma^{0}=\psi_{R}^{\dagger}$$ and likewise for the right handed Weyl spinor? This is the only definition that seems to make sense to me but I'd like to check to make sure.

Given a (Dirac), spinor in the Weyl basis, $\psi = \begin{pmatrix} \psi_{L}\\ \psi_{R} \end{pmatrix} $ , where $\psi_{L}$ and $\psi_{R}$ are Weyl spinors we define the adjoint of the Dirac spinor as;

$\bar{\psi}=\psi^{\dagger}\gamma^{0}=(\psi_{R}^{\dagger},\psi_{L}^{\dagger})$

I understand this, but recently I've ran into expressions like $\bar{\psi}_{L}$, and $\bar{\psi}_{R}$. I can't seem to find a definition for the adjoint of a Weyl spinor so this is confusing me. Is it as simple as; $$\bar{\psi}_{L}=P_{L}\bar{\psi}=P_{L}\psi^{\dagger}\gamma^{0}=\psi_{R}^{\dagger}$$ and likewise for the right handed Weyl spinor? This is the only definition that seems to make sense to me but I'd like to check to make sure.

Edit: In trying to make sense of this I've gone and confused myself further. Another suitable defintion seems to be;

$\bar{\psi}_{L}=\bar{P_{L}\psi}=(P_{L}\psi)^{\dagger}\gamma^{0}=(\psi_{L}^{\dagger},0)\gamma^{0}=(0,\psi_{L}^{\dagger})$ Then taking $\bar{\psi}_{L}=\psi_{L}^{\dagger}$. Which is contrary to my previous idea.

I would really appreciate some clarification on this, thanks.