Here is the definition that I am working on for an adjoint pair of functors:
Def: Let $C,D$ be $n$-categories. A pair of adjoint functors between $C$ and $D$ is pair $$G:C\rightleftarrows D:F$$ of functors and a pair of functor morphisms $$i:1_C\to F\circ G;\quad \text{and}\quad \xi:G\circ F\to 1_D,$$ such that for all $X\in \text{Ob}(C)$ and $Y\in\text{Ob}(D)$, the induced morphisms $$ \hom_C(X,F(Y))\rightleftarrows\hom_D(G(X),Y)$$ are equivalences of $(n-1)$-categories.
I know that if the functor morphisms are natural transformations, then the induced morphisms $\alpha:\hom_C(X,F(Y))\to\hom_D(G(X),Y)$ and $\beta:\hom_D(G(X),Y)\to\hom_C(X,F(Y))$ are given by $$f\mapsto F(f)\circ i_X\quad \text{and} \quad f\mapsto \xi_Y \circ G(f),$$ respectively. So, is functor morphism just another word for natural transformation? Also, the morphisms $\alpha$ and $\beta$ are required to be equivalences of $(n-1)$-categories, so I figure that they should be functors, thus for all $f,g\in \hom_C(X,F(Y))$ and each $2$-morphism $\Gamma\in \hom_C(f,g)$ we have a $2$-morphism $\alpha(\Gamma)\in \hom_D(\alpha(f),\alpha(g))$. Is there a natural way to describe $\alpha(\Gamma)$?