FLP Impossiblity Result assumption of $C_1 = e'(C_0)$
At least in this part of the article, the character C is used for bivalent configurations. The index $0$ for the bivalent $C_0$ was chosen simply because $e(C_0)$ is 0-valent and the index $1$ in $C_1$ was chosen because $e(C_1)$ is 1-valent, even though $C_0$ and $C_1$ are bivalent neighbors (when applying $e'$).