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We study a categorical generalisation of tree automata, as Σ-algebras for a fixed endofunctor Σ endowed with initial and final states. Under mild assumptions about the base category, we present a general minimisation algorithm for these automata. We build upon and extend an existing generalisation of the Nerode equivalence to a categorical setting, and relate it to the existence of minimal automata. Lastly, we show that generalised types of side-effects, such as non-determinism, can be capturedarXiv:1904.08802v1 fatcat:viekhxtihfcn3jrujycshvroii