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A Lawvere 2-theory is a categorification of Lawvere theory, allowing theories for structured categories, for instance.
Noson S. Yanofsky?, The Syntax of Coherence, Cahiers de Topo. Geom. Diff. Cat. Vol XLI-4 Pgs 255 - 304, 2000. arxiv:9910006
Rory B. B. Lucyshyn-Wright, Enriched algebraic theories and monads for a system of arities, Theory and Applications of Categories, Vol. 31, 2016, No. 5, pp 101-137. arXiv:1511.02920, TAC