We first classify Hopf hypersurfaces of both \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2\times {\mathbb {H}}^2\) which satisfy one of the three conditions: (1) constant mean curvature, (2) constant scalar curvature, (3) constant squared norm of the shape operator. It follows that these three conditions are equivalent for a Hopf hypersurface of both \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2 \times {\mathbb {H}}^2\) . Then, we classify hypersurfaces of both \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2\times {\mathbb {H}}^2\) whose structure Jacobi operator is of Codazzi type. As its direct consequence, we obtain the classification of hypersurfaces in both \({\mathbb {S}}^2\times {\mathbb {S}}^2\) and \({\mathbb {H}}^2\times {\mathbb {H}}^2\) for which the structure Jacobi operator satisfies one of the six conditions: (1) vanishing, (2) parallel, (3) recurrent, (4) semi-parallel, (5) Lie parallel, (6) Killing type.
