Charge conjugation
Definition: Charge conjugation
Charge conjugation is a discrete symmetry transformation, which replaces particles with anti particles, effectively reversign the sign of all charges in a theory.
In QED spinors transform under charge conjugation $C$ as $$C\psi=\psi_c=-i\gamma_2\psi^*$$
Charge conjugation of spinors
One can show that $$\psi_c=-i\gamma_2\psi^*$$ by starting with the Dirac equation $$(i\cancel{\partial}-e\cancel{A}-m)\psi=0.$$ We can conjugate this equation and insert a $\gamma^2_2=-1$ to arrive at $$(i\hat{\gamma}_\mu\partial_\mu + e\hat{\gamma}_\mu A_\mu-m)\psi_c=0,$$ with $\hat{\gamma}\mu=\gamma_2\gamma^*\mu\gamma_2$. This is the Dirac equation with a different representation of the gamma matrices $\hat{\gamma}\mu$. This is equally valid as the $\hat{\gamma}\mu$ satisfy the Dirac algebra. We can see that the sign of the $e\cancel{A}$ term is reversed, which means $\psi_c$ satisfies the Dirac equation with flipped charge.
For Majorana Fermions $\psi = \psi_c$.
For the basis vectors of the Dirac spinor one finds $$(u_\uparrow)^c=v_\downarrow,; (u_\downarrow)^c=v_\uparrow,; (v_\uparrow)^c=u_\downarrow,; (v_\downarrow)^c=u_\uparrow.$$
Since $\gamma^*_2=\gamma_2$ $$C\psi^*=-i\gamma_2\psi$$ and therefore $$C^2=1$$
One can show that $$C(\bar{\psi}\psi)=\bar{\psi}\psi$$ $$C(\bar{\psi}\cancel{\partial}\psi)=\bar{\psi}\cancel{\partial}\psi$$ $$C(\bar{\psi}\gamma^\mu\psi)=-\bar{\psi}\gamma^\mu\psi$$
Therefore the free Dirac Lagrangian is invariant under charge conjucation $C$, but the interaction term $eA_\mu\bar{\psi}\gamma^\mu\psi$ will only be $C$ invariant if $$C(A_\mu)\to -A_\mu$$ The kinetic term of $A$, $F^2_{\mu\nu}$ will be invariant anyways if $A_\mu\to \pm A_\mu$.
We therefore choose the transformation property of $A_\mu$ under $C$ in order to preserve the symmetry of the full interacting Lagrangian.