Let $C$ and $D$ be small categories and let $F:C\rightleftarrows :D$ be a n adjunction between them. Given a 2-functor $J$, it is well known that $J(F):J(C)\rightleftarrows :J(D)$ is also an adjunction.
Let $C$ and $D$ be strict 2-categories and let $F:C\rightleftarrows :D$ be a 2-adjunction between them. If we have a 2-functor $J$, is $J(F):J(C)\rightleftarrows :J(D)$ is also an adjunction? What if $J$ were a 3-functor?
Yes, strict 2-functors preserves strict 2-adjunctions between strict 2-categories. A 2-adjunction is just an adjunction in the 2-category of 2-categories, after all! You would need a 3-functor to preserve pseudo, lax, or colax 2-adjunctions, in which the triangle identities are weakened.