>Then there are categories like \$$\mathbf{FinVect}\$$, the category of finite-dimensional vector spaces and linear maps, that are equivalent to their own opposite. These are neither like \$$\mathbf{Set}\$$ nor like \$$\mathbf{Set}^{\text{op}}\$$, but somehow poised right in between.

WOW. I knew vector spaces are beautiful mathematical objects but this is yet again mind blowing. Sounds like they are kind of like a hologram of adjunctions.