Definition 2.4 (Representable). We say a functor $F:C\to Set$ is representable if it’s isomorphic to a $C(A,\bullet )$ for some $A$. By a representation of $F$, we mean a pair $(A,x)$ where $x\in FA$ is such that $\mathrm{\Phi }(x)$ is an isomorphism. We call $x$ a universal element of $F$.