Overcategories and free monoids for overcategories

An overcategory with base category C is merely any functor into C. In this paper we extend the work of Dominique Bourn and Jacques Penon (“Cat'egorification de structures d'efinies par monade cart'esienne”) on overcategories. In particular we show that Freyd’s adjoint theorem, a theorem of Barr and Wells (“Toposes, Triples and Theories”), all remain true in the context of overcategories. We also show that a free monoid construction remains valid in the context of overcategories. The motivation for this study is the development of higher categories as found in the work of Dominique Bourn and Jacques Penon (“Cat'egorification de structures d'efinies par monade cart'esienne”).