This page describes some nice properties of the category of cocommutative coalgebras over a ground ring , in particular details of the proof that it is a complete, cocomplete, lextensive, cartesian closed category with a generating set.
Michael Barr, Coalgebras over a commutative ring, J. Alg. 32 (1974), 600–610.
Hans-Eberhard Porst, On corings and comodules, Archivum Mathematicum (2006), No. 4, 419-425.