What is the relationship between self-duality and groupoid-ness? Does any condition imply the other? Is there an example which helps understand the difference?

To go from a self-duality $F$ on a category to a groupoid, I guess I have to check for $f: A \to B$ that $F(f) \circ f$ and $f \circ F(f)$ are the identities on $A$ and $B$ respectively. I also guess it is safe to assume that all groupoid are isomorphic (not just equivalent) to its dual.

Am I right in my guesswork?