Representable functor

Remarks

A representation $\Phi \colon \Hom(A, -) \cong F$ is determined by the element $u = \Phi_A(\id_A) \in F(A)$, called a universal element. The pair $(A, u)$ represents $F$ exactly when for every object $X$ and every $x \in F(X)$ there is a unique $f \colon A \to X$ with $F(f)(u) = x$.

Example: the contravariant power set functor $P \colon \mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ sends $X$ to its set of subsets and $f \colon X \to Y$ to the preimage map $S \mapsto f^{-1}(S)$. It is represented by $A = \{0, 1\}$: a function $\chi \colon X \to \{0, 1\}$ corresponds to the subset $\chi^{-1}(1)$, and this bijection commutes with preimages.

Non-example: the constant functor $F \colon \mathbf{Set} \to \mathbf{Set}$ with value the empty set is not representable. For every set $A$ the set $\Hom(A, A)$ contains $\id_A$, so it is not in bijection with $F(A) = \emptyset$.