structural:reformatting, information:extraction, linguistic:translation
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 J {\displaystyle J}). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph G {\displaystyle G}. If we let J {\displaystyle J} be the free category generated by G {\displaystyle G}, there is a universal diagram F: J → C {\displaystyle F:J\to C} whose image contains G {\displaystyle G}. 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.