I have a category $\mathcal{C}$ and a contravariant functor $F:\mathcal{C}\to\mathcal{C}$, so that $F^2$ and $\text{id}_{\mathcal{C}}$ are covariant functors $\mathcal{C}\to\mathcal{C}$. I also have a natural isomorphism $\eta:\text{id}_{\mathcal{C}}\to F^2$.
What would be the most accurate/widely accepted way to refer to such $F$? A 'contravariant involution', a 'duality of $\mathcal{C}$ with itself', a 'self-duality of $\mathcal{C}$'?
According to https://mathoverflow.net/questions/92355, self-duality requires additionally that the natural isomorphism is a unit of a self-adjunction of the contravariant functor, i.e. that the zig-zag idenity $F\eta_A\circ \eta_{FA}\colon FA\to FFFA\to FA$ holds for all $A$.
Lacking that condition, contravariant involution would be the better term.