Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres"-also objects of the same category-subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the decalage comonad Don small categories. A simple case involves unaryoperadic categories-ones wherein each map has exactly one abstract fibre-which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad (D) over tilde induced on Cat(D) by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified decalage" comonad D-m on the arrow category Cat(2). (C) 2020 Elsevier Inc. All rights reserved.