Product Category
The product of two categories
and , written , has objects of the form for and , and morphisms of the form for and . Composition and units are defined componentwise. That is There are two obvious projection functors: defined by and , and similarly for .
Diagonal Functor
Define the diagonal functor
such that
Prop For any
Opposite Category
The opposite or dual category
of a category has the same objects as , and an arrow in is an arrow in . That is is just with all of the arrows formally turned around. It is convenient to have a notation to distinguish an object in from the same one in . Thus we may write in for in . With this notation we can define composition and units in in terms of the corresponding operations in , namely
e.g. Consider a simple 
Arrow Category
The arrow category
of a category has the arrows of as objects, and an arrow from to in is a commutative square: where
and are arrows in . That is, such an arrow is a pair of arrows in such that . The identity arrow on an object is the pair . Composition of arrows is done componentwise:
Prop Observe that there is a functor
Slice Category
The slice category
of a category over an object has all arrows such that as objects and from to is an arrow in such that , as indicated in: with identity arrow
and composition law inherits from .
Definition
Def Coslice Category The coslice category
of a category under an object has as objects all arrows of such that , and an arrow from to is an arrow such that : We have identity arrow
and composition law inherits from . e.g. Consider the category of pointed sets consists of sets with a distinguished element , and arrows are functions that preserves the points, . This is isomorphic to the coslice category: where is the singleton. Indeed the functor is an isomorphism.
Prop For any category