Sorry, but I don't really understand this proof:
(paper: "Optimal transport in Lorentzian synthetic spaces, synthetic Ricci curvature lower bounds and applications",[1]: https://arxiv.org/pdf/2004.08934.pdf, Proposition 2.32, point 6)
My principal doubt is: in (2.26), why can we talk about curves that intersect at $t_0$? Does this come from the fact that the measures $\bar{\eta},\bar{\eta}^i$, $i=1,2$, are concentrated on $e^{-1}_{t_0}(A)$ by construction? Why the combination of (2.25) and (2.26) gives the equality of the supports?
And also, if $e_t:C([0,1],X)\rightarrow X$ is the evaluation map at time $t$, $e^{-1}_t(A)$ is the set of curves that "create" $A$ by passing through its points at time $t$?