Universal arrow
Let be a functor. For the objects , the universal arrow from to is defined as the pair with object and morphism :
Said differently, every arrow factors uniquely through .
Examples:

is defined as the vector space over field . The forgetful functor is a function sending a vector space to its underlying elements. For any set , there exists a with as the basis. This fact is illustrated by the function , and there also exists obtained by linear transformation. This makes a universal functor.

Free category over graph is given by the universal functor . Similarly, we have universal functors for a free group or free Rmodule on a given set of generators.

For integral domain and field of quotients , there exists a monomorphism . The universal (forgetful) functor maps , from the category of fields to the category of domains, provided we take the arrows of to be monomorphisms of integral domains.
A universal element of functor is defined to be the pair consisting of , , such that consisting of and arrow implies that .
Consider the equivalence relation on set , and the projection sending every element in the set to its Eequivalence class:
This states that is a universal element under the functor .
, the morphism that sends each group to its quotient group, is another example of the universal element for the functor .
Relationship between universal arrow and universal element: An element can be considered universal arrow from the onepoint set ; then from to is a universal element. If is a functor, then the pair is a universal arrow if is a universal element of the functor .
Projections and give us another example of universal arrows. Given any other pair of arrows , , and , we get making a universal pair. To make it a universal arrow, consider the diagonal functor so that the pair ; then is a universal arrow from to .
Yoneda lemma
The concept of universality can be formulated in terms of homsets as follows. Given functor , the pair for is universal if and only if the function and the composition satisfies the bijection of homsets . This is natural in . If have small homsets, then the functor to is isomorphic to the covariant homfunctor . Such isomorphisms are called representations.
Let be a category with small homsets. Then, representation of functor , given , is the pair such that:
is called representing object, and is termed representable, when such a representation exists. A representation is hence just a covariant homfunctor .
Let be a category with small homsets, , and denote the onepoint set. If is a universal arrow from to , then function which, for every object of , sends the arrow to is a representation of .
A universal arrow from to can be written as the natural isomorphism , or equivalently as representation of the functor , or equally well as the universal element of the same functor.
Informally speaking, the Yoneda lemma states that an object can be recovered by knowing the maps into it. Formally, given category with small homsets, , functor , there is a bijection:
which sends each NT to , the identity . The proof is indicated as:
The bijection is the natural isomorphism . The object function and the arrow function:
define a fully faithful functor , and this is called the Yoneda functor.
Coproducts and colimits
For any category , the diagonal functor is given by the object mapping and morphism mapping . A universal arrow from object to , given by is called the coproduct diagram. When such a diagram exists, object is written as :
The assignment is a bijection that is natural in :
Examples include the disjoint union in , wedge product in , and the tensor product in .
For infinite coproducts, replace with , for any set :
If for all x, copower is written as :
If has a null object such that is the zero arrow, cokernel of is given by :
Given a pair of arrows , coequalizer can be defined as:
Let represent a category with two objects and two nonidentity arrows between them. Then, functor category can be formed:
Having defined the diagonal functor , and object , consider:
In other words, coequalizes . A coequalizer of pair is simply a universal arrow .
As an example, coequalizer in for the set of functions is simply the projection where is the least equivalence relation .
Given , pushout is defined as the pair such that:
The coproduct over , also called cocartesian square or fibered sum is written as:
The pushout of always exists in ; it is the disjoint union with elements identified with and , for .
For , the cokernel pair is defined as the pair of pushout of along with . Indeed, there is some so that are parallel arrows:
Let be a category, and be an index category. The diagonal functor
sends each object to the constant functor  the functor which has the same value for each object , and the value at each arrow of . Given , is an NT. Arrows correspond to objects of . The universal arrow from to is called a colimit (or direct limit, inductive limit) diagram for ; it consits of an object , usually written , or , along with an NT which is universal among NTs . Pictorially, the following diagram commutes:
is often written as , and is called the cone of to the base :
Alternatively, colimit of consists of object along with the cone , from base to the vertex , which is a universal cone (or limiting cone).
As an example, consider , and functor which maps every arrow in to an inclusion map (subset in set). This functor is simply the inclusion . The union of all sets, with cone given by inclusion map , is . For small, any has a colimit.
Limits and products
The notion of a limit is dual to that of a colimit. Let be a category, be an index set, and be a diagonal functor. Limit of functor is defined as the universal arrow from to ; it consists of object , , called the limit object (or projective limit, inverse limit) of , and the NT , which is universal among for . is then called cone to base from vertex , pictured as:
The properties of limits and colimits may be pictured as:
For a discrete category and category , limit of functor consisting of pairs , is called product of and written as . Then, we have projections defined as:
They form a cone with vertex , and we have the bijection of sets natural in :
We then define , and are called the components of . In , this corresponds to direct product.
Infinite products. When is a set (= discrete category, category with all arrows identities), is simply a indexed family of objects , while cone with vertex and base is a indexed family of arrows . We have and the following bijection of sets natural in :
Products over any small set exist in , , and ; they are simply cartesian products.
If factors in a product are all equal, for all , then is called power, and is written as . The following bijection of sets is natural in :
Given and functor defined by parallel arrows , the limit point , when it exists, is called an equalizer or difference kernel of and :
The limit arrow amounts to cone from vertex . In , equalizer always exists; is the set , and is an injection of this subset of into . In , the equalizer is kernel of difference homomorphism . Any equalizer is necessarily a monic.
For , functor is a pair of arrows . The cone of such a functor is a pair of arrows from vertex in the following diagram:
The diagram also illustrates the universal cone formed by , and the square is called pullback square. The product is then called pullback or fibered product.
In , if is a "fiber map" with base , and is a continuous map into the base, then projection of the pullback is the induced fiber map. When it exists, the pullback of a pair of equal arrows is called the kernel pair of . A limit of the empty functor in is a terminal object of .
Categories with finite products
A category is said to have finite products if, for any finite number of objects , there exists a product object and projections for , with the usual universal property.
If has a terminal object and products for any two objects , then has all finite products. The products provide, by , the bifunctor . For any three objects, we have the isomorphism natural in :
For any object , there are isomorphisms natural in :
The proof can be expressed as a commutative diagram:
The dual result holds for a coproduct. A coproduct diagram consists of injections and a map determined by its cocomponents for . If has both finite products and finite coproducts, the arrows
from coproducts to products is determined uniquely by an matrix of arrows . In any category with a null object and the zero arrow through , finite coproducts, and finite products, there is a canonical arrow of the coproduct to the product:
This arrow is precisely the identity matrix of order . It may be an isomorphism (in ), a proper monic (in ), or a proper epi (in ).
Groups in categories
Let be a category with finite products and terminal object . Then, a monoid is defined by the triple such that the following diagrams commute:
where is the associativity morphism. A group is then defined by the triple together with the operation such that the following diagram commutes (with the diagonal):
This suggests that sends each element to its right inverse. One can draw similar diagrams for any algebraic system.
If is a category with finite products, then object is a group iff the homfunctor is a group in the functor category . Each multiplication of determins a corresponding multiplication in the homset as the composite
where , while the first natural isomorphism is given by the definition of the product object . Conversely, given any natural , the Yoneda lemma proves that there is a unique such that . A diagram chase shows that is associative only if is.
Colimits of representatable functors
Any functor , from a small category to the category of Sets, can be represented, in a canoical way, by the colimit of a diagram of representable functors , where . First, given , we construct a diagram category for the colimit , the "category of elements" of ; that is the comma category of object pairs , with as objects, and arrows , those arrows for which . Then, is the colimit of the diagram on given by the functor
which sends each object to the homfunctor , and each arrow to the induced NT . Then, the Yoneda isomorphism on yields a cone over base to :
and we get the following diagram:
This cone to is colimiting over . Consider any other cone from to the vertex ; by the Yoneda lemma, this is an NT , given by , as well as by , for some . To show that the cone to is universal, construct an NT and set . Since , is natural.
For small category , the contravariant functor is called a presheaf, and the category of these functors is written as . This terminology comes from the case when is a category of open subsets of a topological space, and smooth functions; then, for the inclusion , we have the map .