The following covariant functors are representable.
- The identity functor $1_\textup{\textsf{cat}} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ is represented by the singleton set $1$. That is, for any set $X$, there is a natural isomorphism $\textup{\textsf{cat}}(1,X) \cong X$ that defines a bijection between elements $x \in X$ and functions $x \colon 1 \to X$ carrying the singleton element to $x$. Naturality says that for any $f \colon X \to Y$, the diagram
\(\xymatrix{ \cat{Set}(1,X) \ar[d]_{f_*} \ar[r]^-\cong & X \ar[d]^f \\ \textup{\textsf{cat}}(1,Y) \ar[r]_-\cong & Y}\) commutes, i.e., that the composite function $1 \xrightarrow{x} X \xrightarrow{f} Y$ corresponds to the element $f(x) \in Y$, as is evidently the case.
- The forgetful functor $U \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ is represented by the group $\mathbb{Z}$. That is, for any group $G$, there is a natural isomorphism $\textup{\textsf{cat}}(\mathbb{Z},G) \cong UG$ that associates, to every element $g \in UG$, the unique homomorphism $\mathbb{Z} \to G$ that maps the integer 1 to $g$. This defines a bijection because every homomorphism $\mathbb{Z} \to G$ is determined by the image of the generator $1$; that is to say, $\mathbb{Z}$ is the free group on a single generator. This bijection is natural because the composite group homomorphism $\mathbb{Z} \xrightarrow{g} G \xrightarrow{\phi} H$ carries the integer 1 to $\phi(g) \in H$.
- For any unital ring $R$, the forgetful functor $U \colon \textup{\textsf{cat}}_R \to \textup{\textsf{cat}}$ is represented by the $R$-module $R$. That is, there is a natural bijection between $R$-module homomorphisms $R \to M$ and elements of the underlying set of $M$, in which $m \in UM$ is associated to the unique $R$-module homomorphism that carries the multiplicative identity of $R$ to $m$; this is to say, $R$ is the free $R$-module on a single generator. This explains the appearance of the abelian group $\mathbb{Z}$ and the vector space $\mathbbe{k}$ in Definition \ref{defn:monoid-cases}, where maps with these domains were used to specify elements in the codomains.
- The functor $U \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ is represented by the unital ring $\mathbb{Z}[x]$, the polynomial ring in one variable with integer coefficients. A unital ring homomorphism $\mathbb{Z}[x] \to R$ is uniquely determined by the image of $x$; put another way, $\mathbb{Z}[x]$ is the free unital ring on a single generator.
- The functor $U(-)^n \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that sends a group $G$ to the set of $n$-tuples of elements of $G$ is represented by the free group $F_n$ on $n$ generators. Similarly, the functor $U(-)^n \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ is represented by the free abelian group $\Directsum_n \mathbb{Z}$ on $n$ generators.
- More generally, any group presentation, such as
\(S_3 \coloneqq \Big\langle s,t \Bigm\mid s^2=t^2=1, sts=tst \Big\rangle,\) defines a functor $\textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that carries a group $G$ to the set \(\Big\{ (g_1, g_2) \in G^2 \Bigm\mid g_1^2 = g_2^2 =e, g_1g_2g_1 = g_2g_1g_2 \Big\}.\) The functor is represented by the group admitting the given presentation, in this case by the symmetric group $S_3$ on three elements: the presentation tells us that homomorphisms $S_3 \to G$ are classified by pairs of elements $g_1,g_2 \in G$ satisfying the listed relations.
- The functor $(-)^\times \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that sends a unital ring to its set of units is represented by the ring $\mathbb{Z}[x,x^{-1}]$ of Laurent polynomials in one variable. That is to say, a ring homomorphism $\mathbb{Z}[x,x^{-1}] \to R$ may be defined by sending $x$ to any unit of $R$ and is completely determined by this assignment, and moreover there are no ring homomorphisms that carry $x$ to a non-unit.
- The forgetful functor $U \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ is represented by the singleton space: there is a natural bijection between elements of a topological space and continuous functions from the one-point space.
- The functor $\mathrm{ob} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that takes a small category to its set of objects is represented by the terminal category $\mathbbe{1}$: a functor $\mathbbe{1} \to \mathsf{C}$ is no more and no less than a choice of object in $\mathsf{C}$.
- The functor $\mathrm{mor} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that takes a small category to its set of morphisms is represented by the category $\mathbbe{2}$: a functor $\mathbbe{2} \to \mathsf{C}$ is no more and no less than a choice of morphism in $\mathsf{C}$. In this sense, the category $\mathbbe{2}$ is the free or walking arrow.
- The functor $\textup{fun} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that takes a small category to its set of isomorphisms (pointing in a specified direction) is represented by the category $\mathbb{I}$, with two objects and exactly one morphism in each hom-set. In this sense, the category $\mathbb{I}$ is the free or walking isomorphism.
- The functor $\textup{fun} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that takes a small category to the set of composable pairs of morphisms in it is represented by the category $\mathbbe{3}$. Generalizing, the ordinal $\mathbbe{n}+\mathbbe{1} = 0 \to 1 \to \cdots \to n$ represents the functor that takes a small category to the set of paths of $n$ composable morphisms in it.
- The forgetful functor $U \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$ is represented by the two-element based set: based functions out of this set correspond naturally and bijectively to elements of the target based set, the element in question being the image of the non-basepoint element.
- The functor $\textup{fun} \colon \textup{\textsf{cat}} \to \textup{\textsf{cat}}$ that carries a topological space to its set of paths and the functor $\textup{fun} \colon \textup{\textsf{cat}}_* \to \textup{\textsf{cat}}$ that carries a based space to its set of based loops are each representable by definition, by the unit interval $I$ and the based circle $S^1$, respectively. A path in $X$ is a continuous function $I \to X$ while a (based) loop in $X$ is a based continuous function $S^1 \to X$.