Opposite category

The opposite category of a category has the same objects and one reversed morphism for each original morphism, with composition reversed accordingly.

Category: A structure of objects and composable morphisms satisfying identity and associativity laws. The opposite category is built from a category by reversing its morphisms.
Initial object: An object with exactly one morphism to every object in a category. It becomes a terminal object in the opposite category.
Contravariant functor: A map between categories that reverses the direction of morphisms while preserving identities and composition in reverse order. It can be represented as an ordinary functor whose domain is an opposite category.
Opposite group: A group with the same elements as another group and multiplication performed in reverse order. It is the algebraic analogue of reversing morphisms, not a category-level operation.
Morphism: A composable arrow between objects in a category. Each morphism becomes a reversed arrow with its source and target exchanged.
Terminal object: An object with exactly one morphism from every object in a category. It becomes an initial object when all arrows are reversed.
Presheaf: A contravariant functor from a category to sets, assigning data to objects and restriction maps to morphisms. Its contravariance is encoded as a functor from the opposite category.
Product category: A category whose objects and morphisms are pairs drawn from two categories. It combines categories componentwise instead of reversing the arrows of one category.
Composition of morphisms: The operation that combines composable morphisms into a single morphism. The opposite category defines composition in the reverse order.
Product (category theory): An object equipped with projections that is universal among cones into a family of objects. Products in a category correspond to coproducts in its opposite.

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