The forgetful functor $U : \mathbf{Grp} \to \mathbf{Set}$ is representable: $U(G) = \mathrm{Hom}_{\mathbf{Grp}}(\mathbb{Z}, G)$ since a group homomorphism $\mathbb{Z} \to G$ is determined by the image of $1$, giving a bijection with $U(G)$. The representing object is $\mathbb{Z}$ and the universal element is $1 \in U(\mathbb{Z})$.

For a ring $R$, the forgetful functor $U : R\text{-}\mathbf{Mod} \to \mathbf{Set}$ is representable: $U(M) \cong \mathrm{Hom}_R(R, M)$, since an $R$-module homomorphism $R \to M$ is determined by the image of $1 \in R$. The representing object is $R$ itself (viewed as a left module).

For a set $S$ and a fixed element $s \in S$, the evaluation functor $\mathrm{ev}_s : \mathbf{Set}^S \to \mathbf{Set}$ sending $f \mapsto f(s)$ is representable, represented by the characteristic function of $s$.

The product $A \times B$ in a category $\mathcal{C}$ (if it exists) represents the functor $X \mapsto \mathrm{Hom}(X, A) \times \mathrm{Hom}(X, B)$. That is:

$\mathrm{Hom}(X, A \times B) \cong \mathrm{Hom}(X, A) \times \mathrm{Hom}(X, B)$

naturally in $X$. The universal element consists of the two projections $\pi_A : A \times B \to A$ and $\pi_B : A \times B \to B$.

Dually, the coproduct $A \sqcup B$ corepresents $X \mapsto \mathrm{Hom}(A, X) \times \mathrm{Hom}(B, X)$:

$\mathrm{Hom}(A \sqcup B, X) \cong \mathrm{Hom}(A, X) \times \mathrm{Hom}(B, X)$

In $\mathbf{Set}$, $A \sqcup B$ is the disjoint union; in $\mathbf{Ab}$, it is the direct sum $A \oplus B$; in $\mathbf{Grp}$, it is the free product $A * B$.

For $R$-modules $M$ and $N$, the tensor product $M \otimes_R N$ represents the functor of bilinear maps: there is a natural isomorphism

$\mathrm{Hom}_R(M \otimes_R N, P) \cong \mathrm{Bil}_R(M \times N, P)$

The universal bilinear map $M \times N \to M \otimes_R N$ is the universal element.

In Galois theory, for a field extension $K/k$, the fiber functor $\omega : \mathbf{FEt}_k \to \mathbf{FinSet}$ sends a finite etale $k$-algebra $A$ to the set $\mathrm{Hom}_k(A, K)$. When $K = k^{\mathrm{sep}}$, this functor is pro-representable: $\omega(A) \cong \mathrm{Hom}(A, \varinjlim L_i)$ where the colimit runs over finite separable extensions.

In algebraic geometry, the Hilbert functor $\mathrm{Hilb}_{X/S}^P : \mathbf{Sch}/S \to \mathbf{Set}$ sends a scheme $T$ to the set of closed subschemes $Z \subseteq X \times_S T$ flat over $T$ with Hilbert polynomial $P$. When this functor is representable, the representing scheme is the Hilbert scheme $\mathrm{Hilb}_{X/S}^P$.

The functor $\mathrm{Hom}_{\mathbf{CRing}}(A, -) : \mathbf{CRing} \to \mathbf{Set}$ is representable by definition, represented by $A$. Under the equivalence $\mathbf{Aff} \simeq \mathbf{CRing}^{\mathrm{op}}$, this becomes $\mathrm{Hom}_{\mathbf{Sch}}(-, \mathrm{Spec}\, A) : \mathbf{Aff}^{\mathrm{op}} \to \mathbf{Set}$, which is the "functor of points" of $\mathrm{Spec}\, A$.

Every presheaf $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ is a colimit of representables: $F \cong \varinjlim_{(A, x) \in \mathcal{C}/F} h^A$. This is the density theorem (or co-Yoneda lemma). It says that representable presheaves form a "basis" for all presheaves, analogous to how every vector is a linear combination of basis vectors.

The power set functor $\mathcal{P} : \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ sending $S \mapsto \mathcal{P}(S) = 2^S$ is not representable. If it were, there would exist a set $A$ with $\mathcal{P}(S) \cong \mathrm{Hom}(S, A)$ naturally. Setting $S = A$ gives $|\mathcal{P}(A)| = |A^A|$, i.e., $2^{|A|} = |A|^{|A|}$, which fails for all infinite cardinals (and most finite ones).

In the homotopy category of CW-complexes, the Brown representability theorem states that every contravariant functor $F : \mathbf{hCW}^{\mathrm{op}} \to \mathbf{Set}$ satisfying the wedge axiom and the Mayer-Vietoris axiom is representable. This gives the existence of classifying spaces: ordinary cohomology $H^n(-, G)$ is represented by the Eilenberg-Mac Lane space $K(G, n)$.