Representable functor

A functor F : \cat C \to \Set is representable iff it is naturally isomorphic to the fixed-domain Hom-functor $ℎ 𝐶 = 𝖢 (𝐶, -)$ for some object $𝐶 \in 𝖢$ cat

\begin{align*} \eta: h_{C} \Rightarrow F : \cat C \to \Set \end{align*}

whence $(𝐶, 𝜂)$ is called a representation of $𝐹$ .