Suppose we have two representables $H_A, H_{A'}: \mathscr A\to \mathbf {Set}$ and a natural transformation $H_A\to H_{A'}$ with components $\alpha_B:H_A(B)\to H_{A'}(B)$. It can be shown that each $\alpha_B(u)=f\circ u$ for some $f:A\to A'$. How to show that this $f$ is unique?
Suppose $f\circ u=g\circ u$. The natural first thought is to cancel $u$, but it's not necessarily an epimorphism, so there must be some more complicated argument that I can't think of.