Presheaves, covariantly and contravariantly representable functors, and representations

Let $\mathcal{C}$ be a locally small category. A presheaf on $\mathcal{C}$ is a functor $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$, with opposite category as in Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ and $\mathbf{S}\mathbf{e}\mathbf{t}$ as in Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$.

A covariant functor $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is covariantly representable when there are an object $R$ and a natural isomorphism

$\theta :\mathcal{C}(R,-)\stackrel{\cong }{\Rightarrow }F\mathrm{.}$

The pair $(R,\theta )$ is a representation of $F$, and $R$ is a representing object.

A presheaf $P:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is contravariantly representable when there are an object $R$ and a natural isomorphism

$\theta :\mathcal{C}(-,R)\stackrel{\cong }{\Rightarrow }P\mathrm{.}$

The same terms are used for $(R,\theta )$ and $R$. The hom-functors exist by The assignments $\mathcal{C}(a,-)$ and $\mathcal{C}(-,a)$ are functors to $\mathbf{S}\mathbf{e}\mathbf{t}$, and natural transformations and natural isomorphisms have the meanings of Natural transformation and its components and Natural isomorphism. When the variance is clear, representable is used without an adjective.