Abstract: We give a characterization à la Obata for certain families of Kähler manifolds. These results are in the same line as other extensions of the well-known Obata's rigidity theorem from Morio Obata, Certain conditions for a Riemanian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340, like for instance the generalizations in Akhil Ranjan and Gopalkrishnan Santhanam, A generalization of Obata's theorem, J. Geom. Anal. 7 (1997), no. 3, 357-375 and Gopalkrishnan Santhanam, Obata's theorem for Kähler manifolds, Illinois J. Math. 51 (2007), no. 4, 1349-1362. Moreover, we give a complete description of the so-called Kähler doubly-warped product structures whose underlying metric is Einstein.