Notes on some categories related to that of finite totally ordered sets

The purpose of these notes is to collect in one place some facts on the category of finite totally ordered sets and some related categories. More specifically, we collect some results on them which will be useful for the study of iteratedly meta theories of algebra in the style of our work arXiv:1601.00301 (arXiv:1601.00301), which is a kind of higher order universal algebra.