Given strict 2-categories, $A$ and $C$, and strict 2-functors $F:A\to C$ and $U:C\to A$, a strict 2-adjunction is given one of the following two equivalent means:
an isomorphism of categories $C(F a,c)\cong A(a,U c)$ for each object $a$ in $A$ and object $c$ in $C$, which is strict 2-natural both in $a$ and in $c$;
a strict 2-natural 2-transformations of 2-functors unit $\eta : Id_A \to U F$, and counit $\epsilon : F U\to Id_B$, satisfying the triangle identities strictly.
There are more relaxed forms of 2-adjunctions, namely the pseudoadjunctions and biadjunctions, both of which can be considered for 2-functors of either strict or weak 2-categories.