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universal mapping property
1 Introduction.
A strong attribute of categories is a uniform description of many seemingly unrelated concepts from diverse interest areas of mathematics. For example, Cartesian products produce new sets from old just as direct product produce new groups from old and Cartesian products of topologies are again topologies. To make these notions uniform one can provide categorical definitions. Because categories assume as their constituents only abstract objects and morphisms, all the properties of such constructions must be given as properties of morphisms between objects. Those constructions which can be characterized by means of the existence of a unique morphism are generally grouped under the heading of universal mapping properties [1, p. 57]. A precise definition will follow below.
As is common with category theory, there are generally not many theorems that can be proved at a categorical level and set to apply to all universal mapping properties. The exception is a generic proof of uniqueness of objects with a universal mapping property. It should be emphasized that most universal mapping properties are a result of the underlying structure of the categories in question and not provided for abstract reasons. (For concrete examples consider the construction of free groups or free Lie algebras.)
Remark 1.
There are contingents that use the phrase universal property especially in the vernacular. Mac Lane does so in passing but his use is not as a defined term but rather as an intuitive concept to be refined to a strict definition later [3, p. 2]
Because the property applies to the uniqueness of mappings the title universal mapping property remains the choice of most mathematical references in print. Authors interested in shorter titles will sometimes substitute Mac Lane’s concept of universal arrows[3, III.1] – a rigorous definition which will be explained below and captures the spirit of the universal mapping property as presently described. The preferred term of morphism for arrow motivates the modern title universal morphism.
Universal mapping properties arise in such constructions as: direct products and direct sums, free groups, free algebras, product topology, StoneČech compactification, tensor product, exterior algebra, inverse and direct limit, pullbacks and pushouts.
2 Universal Morphisms and the Universal Mapping Property
We follow [3, III.1] for our treatment of universal arrows and universal mapping properties.
Definition 2.
Let $U:\mathcal{D}\to\mathcal{C}$ be a functor from a category $\mathcal{D}$ to a category $\mathcal{C}$, and let $X$ be an object of $\mathcal{C}$. A universal morphism from $X$ to $U$ consists of a pair $(A,\phi)$ where $A$ is an object of $\mathcal{D}$ and $\phi:X\to U(A)$ is a morphism in $\mathcal{C}$, such that the following universal mapping property is satisfied:
Whenever $Y$ is an object of $\mathcal{D}$ and $f\colon X\to U(Y)$ is a morphism in $\mathcal{C}$, then there exists a unique morphism $g:A\to Y$ such that the following diagram commutes.
$\xymatrix{&U(A)\ar[dd]&A\ar@{>}[dd]^{g}\\ X\ar[ur]^{{\phi}}\ar[dr]_{f}&&\\ &U(Y)&Y.}$ Figure 1: A universal morphism from $X$ to $U$
The existence of the morphism g intuitively expresses the fact that $A$ is “general enough”, while the uniqueness of the morphism ensures that $A$ is “not too general”.
One can also consider the categorical dual of the above definition by reversing all the arrows. Let $F:\mathcal{C}\to\mathcal{D}$ be a functor, and let $X$ be an object of $\mathcal{D}$. A couniversal morphism from $F$ to $X$ consists of a pair $(A,\phi)$ where $A$ is an object of $\mathcal{C}$ and $\phi:F(A)\to X$ is a morphism in $\mathcal{D}$, such that the following universal mapping property is satisfied:
Whenever $Y$ is an object of $\mathcal{C}$ and $f\colon F(Y)\to X$ is a morphism in $\mathcal{D}$, then there exists a unique morphism $g\colon Y\to A$ such that the following diagram commutes.
$\xymatrix{Y\ar@{>}[dd]^{g}&F(Y)\ar[dd]^{{F(g)}}\ar[rd]^{f}&\\ &&X.\\ A&F(A)\ar[ur]_{{\phi}}&}$ Figure 2: A couniversal morphism from $F$ to $X$
To avoid ambiguity, some authors may call one of these constructions a universal morphism and the other one a couniversal morphism.
3 Properties
Existence and uniqueness.
Defining a quantity does not guarantee its existence. Given a functor $U$ and an object $X$ as above, there may or may not exist a universal morphism from $X$ to $U$ (or from $U$ to $X$). If, however, a universal morphism $(A,\phi)$ does exists, then it is unique up to a unique isomorphism. That is, if $(A^{{\prime}},\phi^{{\prime}})$ is another such pair then there exists a unique isomorphism $g\colon A\to A^{{\prime}}$ such that $\phi^{{\prime}}=U(g)\phi$. This is easily seen by substituting $(A^{{\prime}},\phi^{{\prime}})$ for $(Y,f)$ in the definition of the universal mapping property.
Equivalent formulations.
The definition of a universal morphism can be rephrased in a variety of ways. Let $U$ be a functor from $\mathcal{D}$ to $\mathcal{C}$, and let $X$ be an object of $\mathcal{C}$. Then the following statements are equivalent

$(A,\phi)$ is a universal morphism from $X$ to $U$;

$(A,\phi)$ is an initial object of the comma category $(X\downarrow U)$;

$(A,\phi)$ is a representable functor of $\operatorname{Hom}_{{\mathcal{C}}}(X,U())$.
The dual statements are also equivalent

$(A,\phi)$ is a universal morphism from $F$ to $X$;

$(A,\phi)$ is a terminal object of the comma category $(F\downarrow X)$;

$(A,\phi)$ is a representable functor—representation of $\operatorname{Hom}_{{\mathcal{C}}}(F(),X)$.
Relation to adjoint functors.
Suppose that $(A_{1},\phi_{1})$ is a universal morphism from $X_{1}$ to $U$ and that $(A_{2},\phi_{2})$ is also a universal morphism from $X_{2}$ to $U$. By the universal mapping property, given any morphism $h\colon X_{1}\to X_{2}$ there exists a unique morphism $g\colon A_{1}\to A_{2}$ such that the following diagram commutes
$\xymatrix{X_{1}\ar[r]^{{\phi_{1}}}\ar[d]_{h}&U(A_{1})\ar[d]_{{U(g)}}&A_{1}\ar@% {>}[d]^{g}\\ X_{2}\ar[r]_{{\phi_{2}}}&U(A_{2})&A_{2}\\ }$ 
If every object $X_{i}$ of $\mathcal{C}$ admits a universal morphism to $U$, then the assignment $X_{i}\mapsto A_{i}$ and $h\mapsto g$ defines a functor $V\colon\mathcal{C}\to\mathcal{D}$. The maps $\phi_{i}$ then define a natural transformation from $1_{{\mathcal{C}}}$ (the identity functor on $\mathcal{C}$) to $UV$. The functors $(V,U)$ are then a pair of adjoint functors, with $V$ leftadjoint to $U$. Similar statements apply to the dual situation of morphisms from $U$. If such morphisms exist for every $X\in\mathcal{C}$ one obtains a functor $V\colon\mathcal{C}\to\mathcal{D}$ which is rightadjoint to $U$.
Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let $F$ and $G$ be a pair of adjoint functors with unit $\eta$ and counit $\epsilon$ (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in $\mathcal{C}$ and $\mathcal{D}$.

For each object $X\in\mathcal{C}$, the pair $(F(X),\eta_{X})$ is a universal morphism from $X$ to $G$. That is, for all $f\colon X\to G(Y)$, there exists a unique $g\colon F(X)\to Y$ for which the diagrams below commute.

Dually, for each object $Y\in\mathcal{D}$, the pair $(G(Y),\epsilon_{Y})$ is a universal morphism from $F$ to $Y$. That is, for all $g\colon F(X)\to Y$ there exists a unique $f\colon X\to G(Y)$ for which the following diagrams commute.
$\xymatrix{&GF(X)\ar[dd]^{{G(g)}}&F(X)\ar[dd]^{{F(f)}}\ar[dr]^{g}\\ X\ar[ur]^{{\eta_{X}}}\ar[dr]_{f}&&&Y\\ &G(Y)&FG(Y)\ar[ur]_{{\epsilon_{Y}}}}$ Figure 3: Universal mapping property of a pair of adjoint functors
Universal constructions are more general than adjoint functor pairs. A universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of $\mathcal{C}$ (equivalently, for every object of $\mathcal{D}$).
4 Examples
Tensor algebras.
Let $\operatorname{\mathit{K}Vect}$ be the category of vector spaces over a field $K$, and let $\operatorname{\mathit{K}Alg}$ be the category of $K$algebras (assumed to be unital and associative). Let $U$ be the forgetful functor which assigns to each algebra its underlying vector space. Given any vector space $V$ over $K$ we can construct the tensor algebra $T(V)$ of $V$. The universal mapping property of the tensor algebra expresses the fact that the pair $(T(V),\iota)$, where $\iota\colon V\to T(V)$ is the natural inclusion map, is a universal morphism from $V$ to $U$. Since this construction works for any vector space $V$, we conclude that $T$ is a functor from $\operatorname{\mathit{K}Vect}$ to $\operatorname{\mathit{K}Alg}$. This functor is leftadjoint to the forgetful functor $U$.
Kernels.
Suppose $\mathcal{C}$ is a category with zero morphisms $0_{{AB}}\in\operatorname{Hom}_{{\mathcal{C}}}(A,B)$ (such as the category of groups) and let $f\colon X\to Y$ be a morphism in $\mathcal{C}$. A kernel of $f$ is any morphism $k\colon K\to X$ such that

the composition $fk$ is the zero morphism from $K$ to $Y$;

given any morphism $k^{{\prime}}\colon K^{{\prime}}\to X$ such that $fk^{{\prime}}$ is the zero morphism, there is a unique morphism $u\colon K^{{\prime}}\to K$ such that $ku=k^{{\prime}}$.
To understand this in the framework of the general setting above, we let $\mathcal{D}$ be the category of morphisms in $\mathcal{C}$. The objects of $\mathcal{D}$ are morphisms $f\colon X\to Y$ in $\mathcal{C}$, and a morphism from $f\colon X\to Y$ to $g\colon S\to T$ is a commutative square whose sides are a pair of morphisms $\alpha\colon X\to S$ and $\beta\colon Y\to T$. Diagramatically,
$\xymatrix{X\ar[r]^{f}\ar[d]_{{\alpha}}&Y\ar[d]^{{\beta}}\\ S\ar[r]^{g}&T}$ 
Let $F\colon\mathcal{C}\to\mathcal{D}$ be the functor that maps an object $K\in\mathcal{C}$ to the zero morphism $0_{{\scriptscriptstyle KK}}\colon K\to K$, and that maps a morphism $r\colon K\to L$ to the trivial square with sides $(r,0_{{KL}})$. Now, let $f$ be an object $\mathcal{D}$ (which is the same thing as a morphism $f\colon X\to Y$ in the category $\mathcal{C}$). A kernel $k\colon K\to X$, if it exists, is the same thing as an object $K\in\mathcal{C}$ and a morphism $(k,0)$ in $\mathcal{D}$ that satisfies the couniversal property expressed by the diagram below. I.e., a kernel is the same thing as a universal morphism from $F$ to $f$.
$\xymatrix{K^{{\prime}}\ar@{>}[dr]_{r}&&K^{{\prime}}\ar[dd]_{{k^{{\prime}}}}% \ar@{>}[dr]^{r}\ar[rr]^{0}&&K^{{\prime}}\ar[dd]^{0}\ar@{>}[dr]^{0}\\ &K&&K\ar[ld]_{k}\ar[rr]_{{0~{}~{}~{}}}&&K\ar[ld]^{0}\\ &&X\ar[rr]_{f}&&Y}$ 
Limits and colimits.
Limits and colimits are important special cases of universal constructions. Let $\mathcal{J}$ and $\mathcal{C}$ be categories with $\mathcal{J}$ small ($\mathcal{J}$ is to be thought of as an index category) and let $\mathcal{C}^{\mathcal{J}}$ be the corresponding functor category of functors from $\mathcal{J}$ to $\mathcal{C}$. The diagonal functor $\Delta\colon\mathcal{C}\to\mathcal{C}^{\mathcal{J}}$ is the functor that maps each object $N\in\mathcal{C}$ to the constant functor $\Delta(N)\colon\mathcal{J}\to\mathcal{C}$ (i.e., $\Delta(N)(X)=N$ for each $X\in\mathcal{J}$.) Given a functor $F\colon\mathcal{J}\to\mathcal{C}$ (thought of as an object in $\mathcal{C}^{\mathcal{J}}$), the limit of $F$, if it exists, is nothing but a universal morphism from $\Delta$ to $F$. Dually, the colimit of $F$ is a couniversal morphism from $F$ to $\Delta$.
5 Motivational remarks.
For the most part, the objects in a category which are constructed in terms of universal mapping properties are not new to the theorist of these categories. For example, topologist long knew that a new topology results from a Cartesian product of topologies. However the general vocabulary allows for easier detection and understanding of functors between different categories. For example, the homology functors take the coproducts – wedge products in topology – to coproducts of modules – direct sums in module categories. Proofs of these facts are in no way facilitated by category theory but the terminology is at least uniform and thus easier to conceptualize and express.
Once one recognizes a certain construction as given by a universal mapping property, one gains several benefits.

Universal mapping properties define objects up to a unique isomorphism. One strategy to prove that two objects are isomorphic is therefore to show that they satisfy the same universal mapping property.

The concrete details of a given construction may be messy, but if the construction satisfies a universal mapping property, one can forget all those details; all there is to know about the construct is already contained in the universal mapping property. Proofs often become short and elegant if the universal mapping property is used rather than the concrete details.

If the universal construction can be carried out for every $X$ in $\mathcal{C}$, then we know that we obtain a functor from $\mathcal{C}$ to $D$. For example, forming kernels is functorial; every commutative square $(\alpha,\beta)$ from the morphism $f$ to the morphism $g$ induces a morphism from the kernel of $f$ to the kernel of $g$.

Furthermore, if such a functor can be formed, it is a right or left adjoint to $U$. But right adjoints commute with limits, and left adjoints commute with colimits! So, looking back at the previous example, we can immediately conclude that the kernel of a product is equal to the product of the kernels.
Universal mapping properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.
References
 1 Thomas W. Hungerford Algebra, SpringerVerlag, New York, (1974).
 2 Paul M. Cohen, Universal Algebra, D.Reidel Publishing, Holland, (1981). ISBN 9027712131.
 3 Saunders Mac Lane Categories for the Working Mathematician 2nd ed. Graduate Texts in Mathematics 5. Springer, (1998). ISBN 0387984038.
Appendix \theappendix Acknowledgements and Notes
This entries was adapted, for the most part, from the Wikipedia entry entitled Universal Property. In turn much of Wikipedia’s entry appears in [3, III.1] were the interested reader is directed for further details.
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