On nearly K\"{a}hler and K\"{a}hler-Codazzi type manifolds

Nearly K"{a}hler and K"{a}hler-Codazzi type manifolds are defined in a very similar way. We prove that nearly K"{a}hler type manifolds have sense just in Hermitian and para-Hermitian contexts, and that K"{a}hler-Codazzi type manifolds reduce to K"{a}hler type manifolds in all the four Hermitian, para-Hermitian, Norden and product Riemannian geometries.