On Structures in Arrow Categories

Abstract: In this article we investigate which categorical structures of a category C are inherited by its arrow category. In particular, we show that a monoidal equivalence between two categories gives rise to a monoidal equivalence between their arrow categories. Moreover, we examine under which circumstances an arrow category is rigid and pivotal. Finally, we derive what the (co)algebra, bialgebra and Hopf algebra objects are in an arrow category.