This may be a stupid question, but I just can't find out the solution.
I'm confusion on how to deal with adjoint pairs, which means a pair of $1$-morphisms $(L,R)$ with two $2$-morphisms $\eta\colon 1\to R\circ L$ and $\epsilon\colon L\circ R\to 1$ satisfies the triangle identities, see here.
In the case $L, R$ are functors, things are wonderful since $(L,R)$ is an adjoint pair iff they are adjoint functors to each other, meaning there is a natural bijection \begin{equation} Hom(L-,-)\cong Hom(-,R-). \end{equation} Then, one can deal with the Hom-sets and everything is fine.
However, I have no idea how to deal adjoint pairs in an arbitrary $2$-category.
For example, when one wants to show the adjoint functor is unique up to unique isomorphism, he/she only need to use the Hom-set statement and follows Yoneda lemma, while I haven't seen any way to prove this simple property for the general adjoint pairs.
So, the question is how to deal with the adjoint pairs in a $2$-category? In particular, are there something like the Hom-set statement for adjoint pairs and propositions like Yoneda lemma?