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A representation of a functor $F$ is $(A,x)$ with $x\in F(A)$ such that $h\mapsto F(h)(x)$ is a bijection $\mathcal{C}(A,B)\to F(B)$ for every $B$. These bijections are natural by functoriality. The element $x$ is its universal element.

If $(B,y)$ is another representation, universality gives unique arrows $f:A\to B$ and $g:B\to A$ satisfying $F(f)x=y$ and $F(g)y=x$. Consequently $F(gf)x=x=F(1_A)x$, so $gf=1_A$ by injectivity of the representing bijection. Similarly $fg=1_B$. Therefore $f$ is the unique compatible isomorphism. This proves uniqueness of functor representations; uniqueness refers to an isomorphism carrying the specified universal elements to one another, rather than to every isomorphism between the underlying objects.