1.8Universal Properties

Updated 11 May 2026Chapter (PDF)

The idea behind universal properties is that they generalize specific constructions by emulating their behavior.

Consider $\emptyset$ in the category $\Set$ . Note that, for any set $S \in \Obj(\Set)$ , there is precisely one set-function/morphism $f: \emptyset \to S$ (the empty function). There is no choice to be made for how to map $\emptyset$ to $S$ , obviously.

Conversely, consider singletons of the form $\{*\}$ in the category $\Set$ . Note that, for any set $S \in \Obj(\Set)$ , there is precisely one set-function $f: S \to \{*\}$ (the constant function $f \equiv *$ ). There is no choice to be made for how to map $S$ to $\{*\}$ , obviously.

We generalize this behavior to more general categories using the notion of initial and terminal objects.

[1.8.1]Definition(Initial, Terminal Objects)#

An object $I \in \Obj(\sf{C})$ is said to be initial if, for any $A \in \Obj(\sf{C})$ , there is precisely one morphism that maps $I$ to $A$ . Conversely, an object $T \in \Obj(\sf{C})$ is said to be terminal if, for any $A \in \Obj(\sf{C})$ , there is precisely one morphism that maps $A$ to $I$ . That is,

\forall I \in \Obj(\mathsf{C}): \qquad \Hom_{\sf{C}}(I, A) \text{ is a singleton,}

\forall T \in \Obj(\mathsf{C}): \qquad \Hom_{\sf{C}}(A, T) \text{ is a singleton.}

[1.8.2]Remark(Initial and Terminal Objects of

\Set

Note then that $\emptyset$ is the initial object of $\Set$ and that all singletons $\{*\}$ are terminal objects in $\Set$ . But note that $\emptyset$ is the singular, unique initial object of $\sf{C}$ , while there are infinitely many singletons $\{*\}$ that are terminal objects of $\sf{C}$ . However, we will show that all initial and terminal objects are isomorphic to each other.

[1.8.3]Theorem(Initial and Terminal Objects Are Unique up to Isomorphism)#

Let $\sf{C}$ be a category.

If $I_1, I_2 \in \Obj(\sf{C})$ are initial, then $I_1 \cong I_2$ .
If $T_1, T_2 \in \Obj(\sf{C})$ are terminal, then $T_1 \cong T_2$ .

Proof.

To prove $(1)$ , suppose $I_1$ and $I_2$ are initial objects of $\sf{C}$ . Then note the morphisms $f: I_1 \to I_2$ and $g: I_2 \to I_1$ are unique by the property of $I_1, I_2$ being initial. Note then that $gf \in \End_{\sf{C}}(I_1)$ and $fg \in \End_{\sf{C}}(I_2)$ . But note that morphisms of form $I_1 \to I_1$ and $I_2 \to I_2$ are unique since the two objects are initial, so $|\End_{\sf{C}}(I_1)| = |\End_{\sf{C}}(I_2)| = 1$ . Moreover, these sets must contain an identity morphism, so in reality $fg$ and $gf$ are forced to be identity morphisms. Since $f$ has a two-sided inverse $g$ , it is an isomorphism, completing the proof for $(1)$ . The argument for $(2)$ follows in an entirely symmetric way.

A construction satisfies a universal property if, loosely speaking, it is the initial or terminal object of some category. We describe this by providing intuition for certain constructions satisfying universal properties.

[1.8.4]Remark(Quotient of Set Is Universal)#

Suppose $A$ is some set endowed with an equivalence relation $\sim$ . Consider the following statement:

“ $A/{\sim}$ is universal with respect to the property that $A$ maps to any set $Z$ in such a way that elements equivalent under $\sim$ have the same image.”

To decipher this, we systematically construct an accessory category where this universal construction manifests as an initial or terminal object. In general, whenever a construction is the solution to a universal problem, its accessory category is built as follows:

Objects: The possible “setups” satisfying the described property, generally a set paired with constraining morphisms.
Morphisms: Standard functions between the sets that make the diagrams formed by the constraining morphisms commute.
Initial/Terminal Objects: The universal solution is precisely the initial or terminal object of this category (unique up to isomorphism).

We now design this accessory category for our specific quotient property.

Objects: Pairs $(\phi, Z)$ where $Z$ is a set and $\phi: A \to Z$ is a function satisfying $\phi(a') = \phi(a'')$ whenever $a' \sim a''$ .
Morphisms: A morphism from $(\phi_1, Z_1)$ to $(\phi_2, Z_2)$ is defined as a function $\sigma: Z_1 \to Z_2$ that respects the mappings from $A$ . That is, the following diagram must commute:

Fix some arbitrary object $(\phi, Z)$ . An initial object takes the form $(\bar{\phi}, I)$ such that there exists a unique morphism $\sigma: (\bar{\phi}, I) \to (\phi, Z)$ . Consider the canonical projection $\pi: A \to A/{\sim}$ . Since $a' \sim a'' \implies \pi(a') = \pi(a'')$ , the pair $(\pi, A/{\sim})$ is a valid object. To satisfy the commutative diagram ( $\sigma \circ \pi = \phi$ ), we are forced to define $\sigma([a]) = \phi(a)$ . Because $\phi$ respects $\sim$ , this $\sigma$ is uniquely determined and well-defined. Thus, $(\pi, A/{\sim})$ is indeed initial.

Conversely, a terminal object takes the form $(\bar{\phi}, T)$ such that there exists a unique morphism $\sigma: (\phi, Z) \to (\bar{\phi}, T)$ . It makes sense to conjecture that $\bar{\phi}$ sends every element of $A$ to a constant $*$ , making $T = \{*\}$ . Since $\im(\bar{\phi})$ is a singleton, all equivalent elements of $A$ trivially have the same image, satisfying the object constraint. Moreover, $\sigma: Z \to T$ is unique since there is only one possible function mapping any set $Z$ to a singleton. This unique $\sigma$ trivially satisfies the required commutative diagram ( $\sigma \circ \phi = \bar{\phi}$ ), proving $(\bar{\phi}, T)$ is indeed terminal.

[1.8.5]Remark(Product of Sets is Universal)#

Fix two sets $A, B$ . We seek to show the product $A \times B$ satisfies a universal property. Consider the projections of $A \times B$ onto $A$ and $B$ .

We observe that the set in question is $A \times B$ , constrained by morphisms $\pi_A$ and $\pi_B$ . Let's consider an auxiliary category where sets of the form $(X, f_A, f_B)$ are objects—thus $(A \times B, \pi_A, \pi_B)$ form objects, and we have the following diagram.

$\sigma: X \to A \times B$ must be oriented in this direction to make the diagram commute. Now we simply show $A \times B$ is terminal.

Proof.

Fix any $x \in X$ . Then note $(\pi_A \circ \sigma)(x) = f_A(x)$ and $(\pi_B \circ \sigma)(x) = f_B(x)$ by commutativity of the diagram. Suppose any other function, say $\sigma'$ , was such that $\sigma': X \to A \times B$ and made the diagram commute. But then, for each $x \in X$ , the image under $\sigma$ and $\sigma'$ would be equivalent, and thus $\sigma \equiv \sigma'$ , completing the uniqueness proof.

We sum this up by saying the product of two sets $A \times B$ is universal with respect to the two projection maps onto $A$ and $B$ .

[1.8.6]Remark(Disjoint Union of Sets is a Coproduct)#

The coproduct is the dual of the product, where dualization acts by flipping all the arrows in a category. It is trivial to show the disjoint union in $\Set$ is an example of the coproduct.

Previously, we defined the Cartesian product and disjoint union by set-theoretic means and ran into the problem of these definitions to identify unique sets up to isomorphism. With category theory, we have defined these concepts not as sets, but by how they relate to their constituents and how they solve a universal problem. This is a perfectly valid definition that does not have to resolve this ambiguity.