In denotational semantics, we have a few examples of notions which are easy to describe graphically (and generally informally), but whose algebraic axiomatisation is tedious, to say the least. Such examples include (in order of appearance) Lambek's polycategories, Girard's proof nets, and Lafont's interaction nets. The importance of algebraic axiomatisation of course resides in the induced notion of denotational model. In this work, we propose a generalisation of Joyal's analytic functors to certain presheaf categories, which we then use to directly derive algebraic axiomatisations from elementary graphical information. For concreteness, we instantiate our results on polycategories. The idea is that the pictures generally used to describe the standard operations on polycategories may be understood as a straightforward collection of finite presheaves on a certain category. Our notion of analytic functor then allows us to interpret this collection as an endofunctor on presheaves, which then freely generates a certain monad, whose algebras are precisely polycategories.