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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.
If A is an object of C, then the functor from C to Set that sends X to Hom C (X,A) (the set of morphisms in C from X to A) is an example of such a functor. If C is a small category (i.e. the collection of its objects forms a set), then the contravariant functors from C to Set, together with natural transformations as morphisms, form a new ...
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties. If F : C → D is an equivalence, then the following statements are all true: the object c of C is an initial object (or terminal object, or zero object), if and only if Fc is an initial object (or terminal object, or zero object) of D
Let T, η, μ be a monad over a category C.The Kleisli category of C is the category C T whose objects and morphisms are given by = (), (,) = (,).That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in C T (but with codomain Y).
Let C be a category with finite products and a terminal object 1. A list object over an object A of C is: an object L A, a morphism o A : 1 → L A, and; a morphism s A : A × L A → L A; such that for any object B of C with maps b : 1 → B and t : A × B → B, there exists a unique f : L A → B such that the following diagram commutes:
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Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property. [2] Universal properties occur everywhere in mathematics.
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