Higher-order recursion schemes are recursive equations defining new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in lambda-calculus by an endofunctor Hlambda and they explained simultaneous substitution of lambda-terms by proving that the presheaf of lambda-terms is an initial Hlambda-monoid. Here we work with the presheaf of rational infinite lambda-terms and prove that this is an initial iterative Hlambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid.
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