Limits and colimits are defined as universal cones. That is, cones through which all other cones factor. A cone φ from L to F is a universal cone if for any other cone ψ from N to F there is a unique morphism from ψ to φ.
Equivalently, a universal cone to F is a universal morphism from Δ to F (thought of as an object in CJ), or a terminal object in (Δ ↓ F).
Dually, a cone φ from F to L is a universal cone if for any other cone ψ from F to N there is a unique morphism from φ to ψ.
Equivalently, a universal cone from F is a universal morphism from F to Δ, or an initial object in (F ↓ Δ).
The limit of F is a universal cone to F, and the colimit is a universal cone from F. As with all universal constructions, universal cones are not guaranteed to exist for all diagrams F, but if they do exist they are unique up to a unique isomorphism.
Read more about this topic: Cone (category Theory)
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