are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here: A co-cone of a diagram is an object of together with a family of morphisms for every object of , such that for every morphism in , we have . A colimit of a diagram is a co-cone of such that for any other co-cone of there exists a unique morphism such that for all in . Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from . As with limits, if a diagram has a colimit then this colimit is unique up to a unique isomorphism. Variations Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in ). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph . If we let be the free category generated by , there is a universal diagram whose image contains . The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms. Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped. Examples Limits The definition of limits is general enough to subsume several constructions useful in practical settings. In the following we will consider the limit (L, φ) of a diagram F : J → C. Terminal objects. If J is the empty category there is only one diagram of shape J: the empty one (similar to the empty function in set theory). A cone to the empty diagram is essentially just an object of C. The limit of F is any object that is uniquely factored through by every other object. This is just the definition of a terminal object. Products. If J is a discrete category then a diagram F is essentially nothing but a family of objects of C, indexed by J. The limit L of F is called the product of these objects. The cone φ consists of a family of morphisms φX : L → F(X) called