Quasi-Gröbner categories were introduced by Sam and Snowden to unify treatment of categories in representation stability. We give new examples of quasi-Gröbner categories. Most of these categories are operadic categories of Batanin and Markl which are used to encode homotopy coherent structures. This suggests that other operadic categories might also be quasi-Gröbner. Additionally, we show that set-operads form a full subcategory of the category of operadic categories. We state several open problems.