Initial and terminal objects

An initial object has exactly one morphism to every object in its category; a terminal object has exactly one morphism from every object. Either may fail to exist, and each is unique up to isomorphism when it exists.

Connect

Category: A structure consisting of objects and composable morphisms, with identity morphisms and associative composition. Initiality and terminality are defined relative to a category’s objects and morphisms.
Empty set: The set with no elements. It is initial in the category of sets because exactly one function leaves it for each set.
Universal property: A characterization of an object through unique morphisms relating it to other objects. Initial and terminal objects are defined entirely by universal mapping properties.
Monomorphism: A morphism that can be cancelled on the left in composition. It describes a cancellation property of one morphism, unlike initiality’s condition on all outgoing morphisms.
Duality (category theory): A principle that reverses arrows to turn categorical statements into dual statements. Reversing arrows exchanges initial objects with terminal objects.
Morphism: A structure-preserving arrow between objects in a category. The unique arrows to or from an object are what its initial or terminal property specifies.
Singleton set: A set containing exactly one element. It is terminal in the category of sets because each set has exactly one function to it.
Uniqueness up to isomorphism: A mathematical principle that objects satisfying the same universal characterization are uniquely isomorphic. It makes the choice of an initial or terminal object immaterial up to unique isomorphism.
Epimorphism: A morphism that can be cancelled on the right in composition. It concerns cancellation, not the unique incoming morphisms defining terminality.
Opposite category: A category formed by reversing every morphism of another category. An initial object becomes terminal when the category is replaced by its opposite.