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For example, the initial object in any concrete category with free objects will be the free object generated by the empty set (since the free functor, being left adjoint to the forgetful functor to Set, preserves colimits). Initial and terminal objects may also be characterized in terms of universal properties and adjoint functors.
Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see § Connection with comma categories, below). Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that ...
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is complete and co-complete.
The unit type is the terminal object in the category of types and typed functions. It should not be confused with the zero or empty type, which allows no values and is the initial object in this category. Similarly, the Boolean is the type with two values. The unit type is implemented in most functional programming languages.
Pushouts are equivalent to coproducts and coequalizers (if there is an initial object) in the sense that: Coproducts are a pushout from the initial object, and the coequalizer of f, g : X → Y is the pushout of [f, g] and [1 X, 1 X], so if there are pushouts (and an initial object), then there are coequalizers and coproducts;
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In the case of categories whose objects are sets or which have an underlying set, the identity arrow is the identity mapping from the object to itself. This is the case here. This is the case here. For example, in the category of non-empty sets, the objects are sets and the arrows are mappings from a set to another (or to the same) set.