Recall that in the setting of $1$-categories, given functors $L\colon\mathcal{C}\longrightarrow\mathcal{D}$ and $R\colon\mathcal{D}\longrightarrow\mathcal{C}$, the following conditions are equivalent:

1. We have an adjunction $L\dashv R\colon \mathcal{C}\rightleftarrows\mathcal{D}$;
2. For each category $\mathcal{K}$, we have an adjunction $L_*\dashv R_*\colon\mathsf{Fun}(\mathcal{K},\mathcal{C})\rightleftarrows\mathsf{Fun}(\mathcal{K},\mathcal{D})$, i.e. a natural bijection $$ \mathrm{Nat}(L\circ F,G) \cong \mathrm{Nat}(F,R\circ G). $$
3. For each category $\mathcal{E}$, we have an adjunction $R^*\dashv L^*\colon\mathsf{Fun}(\mathcal{C},\mathcal{E})\rightleftarrows\mathsf{Fun}(\mathcal{D},\mathcal{E})$, i.e. a natural bijection $$ \mathrm{Nat}(F\circ R,G) \cong \mathrm{Nat}(F,G\circ L). $$

Passing now to bicategories, the situation gets more complicated: in place of $\mathrm{Nat}$, we have $\mathsf{LaxNat}$, $\mathsf{OplaxNat}$, $\mathsf{PseudoNat}$, and, if $F$ and $G$ are $2$-functors, $\mathsf{2Nat}$. Moreover, these are now categories, having two different notions of sameness: isomorphism and equivalences. This leads to the consideration of notions such as the following:

Definition. An oplax biadjunction is a pair $L,R\colon\mathcal{C}\rightleftarrows\mathcal{D}$ of lax functors such that, for each bicategory $\mathcal{K}$, we have a pseudonatural equivalence of categories $$ \mathsf{OplaxNat}(L\circ F,G) \overset{\mathrm{e.q.}}{\cong} \mathsf{OplaxNat}(F,R\circ G), $$ i.e. such that the pseudobifunctors $\mathsf{OplaxNat}(L\circ(-),-)$ and $\mathsf{OplaxNat}(-,R\circ(-))$ are pseudonaturally equivalent as objects of $\mathsf{PseudoFun}(\mathcal{K},\mathcal{C})^\mathsf{op}\times\mathsf{PseudoFun}(\mathcal{K},\mathcal{D})$.

This leads to a total of eight different notions of $2$-dimensional adjunctions: "lax biadjunction", "oplax bi", "lax", "oplax", "pseudo bi", "pseudo", "$2$-", and "$2$-bi".

Question: How are these definitions related to the ones involving co/units (as defined in e.g. Section 2 of [Gurski]) and those involving isomorphisms/equivalences of $\mathsf{Hom}$-categories (as in e.g. Chapter 9 of [Fiore])?