(Sorry for the bump, everyone, but I only just saw this question.)
Here's an example similar in feel to the $\operatorname{Dis} \dashv U \dashv \operatorname{Codiss}$ example - so perhaps not of the sort that you were really after.
Let ${\bf Op}_1$ be the category whose objects are (complete) operator spaces (or "quantum/quantized Banach spaces" according to some authors) and whose morphisms are the completely contractive maps. Let ${\bf Ban}_1$ be the category of Banach spaces and contractive (a.k.a. short) linear maps. Then if $U:{\bf Op}_1\to{\bf Ban}_1$ is the forgetful functor, we have adjunctions $\operatorname{MAX} \dashv U \dashv \operatorname{MIN}$.
The left and right adjoints to $U$ are sometimes called the maximal and minimal quantizations, respectively, of a Banach space. (One also sees the terminology of "maximal and miimal operator space structures", but then we wouldn't be able to have the magic word quantum and its important-sounding derivatives.)
See Prop 3.3 of this article by Pestov for a brief mention of left adjoints and MAX.
