A left Kan extension is left adjoint to precomposition, when it exists; a Right Kan extension is right adjoint. For small indexing categories and set-valued functors, both exist by comma-category colimit and limit formulas. These two adjoints give the essential geometric morphism induced by a functor between presheaf indexing categories.
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In category theory, a **Kan extension** is a construction used to generalize the idea of extending functions or functors across categories. More specifically, Kan extensions can be thought of as a way to extend a functor defined on a small category to a functor defined on a larger category, while maintaining certain properties related to limits or colimits. There are two types of Kan extensions: **left Kan extensions** and **right Kan extensions**.