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Adjoint functors
Adjoint functors, also called an adjoint situation, are a pair of functors related by a natural bijection of morphisms, introduced by Daniel M. Kan in 1958.
Comonad
A comonad is a category-theory structure dual to a monad: an endofunctor with a counit and comultiplication, arising from adjunctions and used in sheaf cohomology.
Equivalence of categories
Equivalence of categories is the standard category-theoretic notion of sameness between two categories: a pair of functors going in opposite directions, inverse up to natural isomorphism.
Natural transformation
A natural transformation, also called a morphism of functors, assigns to each object a morphism between two parallel functors' values, compatible with every morphism of the source category.