I'm trying to formulate a statement that would characterize adjoint functors in terms of "universal elements", and I'm trying to make the statement look like Theorem 2.2.5 and Theorem 2.3.6 in Leinster's text. Below is my attempt.
Background/Notation: the (dual version of the) Yoneda lemma gives a natural bijection $$[\mathscr A,\mathbf{Set}](H^A,X)\simeq X(A)$$ given from left to right by $\alpha\mapsto \hat\alpha$ where $\hat\alpha=\alpha_A(1_A)$, and from right to left by $u\mapsto \widetilde u$ where $\widetilde u_B:H^A(B)\to X(B)$ is given by $f\mapsto X(f)(u)$.
Proposition.
Let $F:\mathscr A\to \mathscr B,\ G:\mathscr B\to\mathscr A$ be functors. Let $X: \mathscr A(A,G(-)):\mathscr B\to \mathbf {Set}$.
There is a one-to-one correspondence between
(a) adjunctions between $F$ and $G$ (with $F$ on the left and $G$ on the right);
(b) elements $u\in X(F(A))$ such that for every $B\in\mathscr B$, $\widetilde u_B$ is a bijection.
Did I formulate the proposition correctly? (I'm only interested in the statement, not in its proof; I'll prove it myself.)