  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").
