Suppose that $(X, \leq_X)$ and $(Y, \leq_Y)$ are ordered sets. Let $T:(X, \leq_X) \rightarrow (Y, \leq_Y)$ be an order isomorphism. Is it true that $T$ is a lattice isomorphism?
I have this question because I couldn't follow the explanation here by Josse. Can anyone provide an explanation?