how do contravariant 2-functors preserve adjunctions?

Question

I know that covariant 2-functors preserve adjunctions. Do contravariant 2-functors preserve the order left-right of the adjoints, or do they reverse it?

Answer 1

In a 2-category, there are two different compositions (traditionally called horizontal and vertical) which can be reversed independently. If you reverse just one of the compositions, then the handedness of adjunctions is reversed. For example, for a fixed object $C$, $\mathrm{Hom} (-, C)$ reverses only horizontal composition and will send left adjoints in your 2-category to right adjoints in $\mathfrak{Cat}$.

Of course, if you reverse both compositions, then the handedness of adjunctions is preserved. For example, $[(-)^\mathrm{op}, C]$ preserves left adjoints.