Adjunctions and (co)limits: Limits and colimits

This lecture establishes the relationship between adjunctions and limits/colimits. It proves that left adjoints preserve colimits, and right adjoints preserve limits. The lecture introduces Proposition 1.10, which demonstrates that the image of a colimit under the functor L is also a colimit. It further explores Proposition 1.13, showing that the existence of limits/colimits is related to the presence of adjoints. The lecture concludes by discussing the universal property of colimits and limits, emphasizing the uniqueness of morphisms satisfying specific properties. The instructor illustrates these concepts with diagrams and detailed proofs, highlighting the fundamental connection between adjunctions and (co)limits.