The category of presheaves on a category \( C \) consists of contravariant functors from \( C \) to the category of sets, along with natural transformations between them. This category captures the idea of assigning a set to each object in \( C \) while respecting the morphisms between those objects, providing a framework for working with collections of data that vary according to the structure of \( C \). The concept is closely tied to the Yoneda lemma, which establishes a powerful relationship between presheaves and the objects in the category.
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