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We study a certain type of action of categories on categories and on operads.Using the structure of the categories {\Delta} and {\Omega} governing categoryand operad structures, respectively, we define categories which instead encodethe structure of a category acting on a category, or a category acting on anoperad. We prove that the former has the structure of an elegant Reedycategory, whereas the latter has the structure of a generalized Reedy category.In particular, this approach gives a new way to regard group actions oncategories and on operads.

Reedy categories which encode the notion of category actions