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An overcategory with base category C is merely any functor into C. In thispaper we extend the work of Dominique Bourn and Jacques Penon("Cat\'egorification de structures d\'efinies par monade cart\'esienne") onovercategories. In particular we show that Freyd's adjoint theorem, a theoremof Barr and Wells ("Toposes, Triples and Theories"), all remain true in thecontext of overcategories. We also show that a free monoid construction remainsvalid in the context of overcategories. The motivation for this study is thedevelopment of higher categories as found in the work of Dominique Bourn andJacques Penon ("Cat\'egorification de structures d\'efinies par monadecart\'esienne").

Overcategories and free monoids for overcategories