On page 8 of [1], at the start of the proof of well defined-ness, Chodosh says "if $\phi_{12}(x_{1},x_{2}) + \phi_{23}(x_{2},x_{3}) = \tilde\phi_{12}(x_{1},x_{2}) + \tilde\phi_{23}(x_{2},x_{3})$ as functions, then $$\phi_{12}(x_{1},x_{2}) - \tilde\phi_{12}(x_{1},x_{2}) = \tilde\phi_{23}(x_{2},x_{3}) - \phi_{23}(x_{2},x_{3})$$ which shows that both sides are functions of $x_{2}$ only."
What does the "as functions" mean? Also, it is not obvious to me how both sides are functions of $x_{2}$.
Reference: [1] O. Chodosh (2012) Optimal Transport and Ricci Curvature: Wasserstein Space Over the Interval. arXiv: https://arxiv.org/abs/1105.2883
Assuming the signs problem in the first out-of-text formula is cured, the left hand side does not depend on $x_3$ thus it is the same for the right and vice-versa for $x_1$.