It is well-known that reduced smooth orbifolds and proper effective foliation Lie groupoids form equivalent categories. However, for certain recent lines of research, equivalence of categories is not sufficient. We propose a notion of maps between reduced smooth orbifolds and a definition of a category in terms of marked proper effective étale Lie groupoids such that the arising category of orbifolds is isomorphic (not only equivalent) to this groupoid category.
"The category of reduced orbifolds in local charts." J. Math. Soc. Japan 69 (2) 755 - 800, April, 2017. https://doi.org/10.2969/jmsj/06920755