There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops the theory parallel to the theory of equivalent Categories of modules over algebras, is:

• Morita theorems for Categories of comodules, M. Takeuchi, J. Fac. Sci. Univ. Tokyo, 24, (1977), p. 629-644

(Some older partial results are also cited at the epilogue of the paper).
Lots of the results presented there, have been generalized at a more rescent work, considering equivalences of comodule Categories for coalgebras over rings:

• Equivalences of comodule categories for coalgebras over rings, K. Al-Takhman, Journal of Pure and Applied Algebra, 173, (2002), p. 245 – 271

There have been some classic papers, on the development of a Morita theory for equivalent Categories of comodules over coalgebras. I believe one of the oldest and most complete works, which develops the theory parallel to the theory of equivalent Categories of modules over algebras, is:

• Morita theorems for Categories of comodules, M. Takeuchi, J. Fac. Sci. Univ. Tokyo, 24, (1977), p. 629-644

(Some older partial results are also cited in the epilogue of the paper).
Lots of the results presented there, have been generalized at a more rescent work, considering equivalences of comodule Categories for coalgebras over rings:

• Equivalences of comodule categories for coalgebras over rings, K. Al-Takhman, Journal of Pure and Applied Algebra, 173, (2002), p. 245 – 271