I've heard that if $U : \mathbf{C} \rightarrow \mathbf{S}$ and $F : \mathbf{S} \rightarrow \mathbf{C}$ are functors between categories, then there's at most one natural isomorphism $$\mathbf{C}(F-,-) \cong \mathbf{S}(-,U-).$$ In other words, $F$ is left-adjoint to $U$ in at most one way.
Question. Is this actually true? If so, how does one prove it? (Please be more specific than "just use the Yoneda Lemma.") If not, what's a counterexample?
When $U = {\bf C} (X, \_)$, the Yoneda lemma implies that $\text{Aut}(U) \cong \text{Aut}(X)$.
Of couse very often this kind of functors have a left adjoint $F$.
For each of this automorphisms of the functor we get a different functor with the same universal property of $U$ with respect to $F$.