Representable Functor

Let $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ be a Presheaf.
The representation of $F$ is a natural isomorphism to the hom-functor should it exist.

$$\theta :\mathcal{C}[-,a]\cong F$$

In Category Theory, a Functor $F:\mathcal{C}\to \mathbf{Set}$ is representable iff
There is an object $a:\mathcal{C}$ s.t. for all object $b:\mathcal{C}$, the elements F(b) are really maps $b\to a$ .
Say $F$ is represented by the object $c$.