Ž . JOURNAL OF ALGEBRA 185, 527543 1996 ARTICLE NO. 0338 Colby Fuller Duality between Coalgebras J. Gomez Torrecillas ´ Departamento de Algebra, Facultad de Ciencias, Uni  ersidad de Granada, E18071 Granada, Spain and C. Nastasescu  ˘ ˘ Facultatea de Matematica, Strasse Academiei 14, R70109 Bucharest, Romania ˘ Communicated by Susan Montgomery Received May 1, 1995 INTRODUCTION  In 8, K. Morita introduced a useful notion of duality between cate- gories of modules, usually called ‘‘Morita duality.’’ He proved that every duality is given by contravariant hom functors defined by a bimodule which is an injective cogenerator for both categories of modules. On the other hand, the equivalences between categories of comodules over a coalgebra   were characterized by M. Takeuchi 12 . In this paper, we study the dualities between categories of comodules. A notion of duality for general Grothendieck categories that seems to extend Morita duality satisfactorily  was introduced by R. R. Colby and K. R. Fuller in 1 . It has been recently   investigated by J. L. Gomez Pardo and P. A. Guil Asensio 3, 4, 6 . ´ Section 1 is devoted to obtaining a complete characterization of Colby Fuller dualities between coalgebras. A coalgebra C over a field k is right semiperfect if the category M C of right C-comodules has enough projectives. If C and D are coalgebras over a field k, then either C and D are left and right semiperfect or there is no Colby Fuller duality between the category of right C-comodules and the category of left D-comodules Ž . Ž. Theorem 1.11 . This, together with Theorem 1.6 2 , shows that there is a *This paper was written while the second author was a Visiting Professor at the University of Almerıa supported by the DGICYT under Grant SAB94-0290. ´ 527 0021-869396 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.