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 pair of 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. Note that this is an ordinary adjunction internal to the 2-category $2Cat$ of Cat-enriched categories, strict 2-functors, and strict 2-natural transformations.
There are also more relaxed forms of 2-adjunction, involving weak 2-categories, weak 2-functors, and/or weak 2-natural transformations; see 2-adjunction.