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Arboreal categories provide an axiomatic framework in which abstract notionsof bisimilarity and back-and-forth games can be defined. They act onextensional categories, typically consisting of relational structures, viaarboreal adjunctions. In the examples, equivalence of structures in variousfragments of infinitary first-order logic (with finitely many variables) can becaptured by transferring bisimilarity along the adjunction.  In most applications, the categories involved are locally finitelypresentable and the adjunctions finitely accessible. Drawing on this insight,we identify the expressive power of this class of adjunctions. We show that theranks of back-and-forth games in the arboreal category are definable byformulae \`a la Hintikka. Thus, the relation between extensional objectsinduced by bisimilarity is always coarser than equivalence in infinitaryfirst-order logic.  Our approach relies on Gabriel-Ulmer duality for locally finitely presentablecategories, and Hodges' word-constructions.

Finitely accessible arboreal adjunctions and Hintikka formulae