Representation of a functor

A representation of $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ together with a universal element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, it is a specified natural isomorphism $\mathcal{C}(A,-)\cong F$.

The universal element in a representation of a functor is the image of $1_A$ under the representing natural isomorphism. Any two universal elements induce unique mutually inverse maps between their representing objects, so representations are unique up to unique compatible isomorphism.