This page on Wikipedia says, "In the language of quantum field theory, charge conjugation transforms as -

• $\psi \Rightarrow -i\big(\bar{\psi} \gamma ^0 \gamma ^2 \big)^T $
• $\bar{\psi} \Rightarrow -i\big(\gamma ^0 \gamma ^2 \psi \big)^T $
  

Could someone explain the reason behind choosing this specific mathematical transformation for charge conjugation (since the only thing it's supposed to do is invert the sign of all charges to convert a particle into its anti-particle)? Any relevant sources are also welcome.