Todd Trimble <trimble1 <at> optonline.net> writes:
The finite-dim cocommutative coalgebras over k coincide with the finitely presentable objects in CocommCoalg_k, and the category of finite-dim cocommutative coalgebras is dual to the category of finite-dim commutative algebras/k. One concludes (a la Gabriel-Ulmer duality) that there is an equivalence
CocommCoalg ~ Lex(CommAlg_{fd}, Set),
where the right side is the category of left exact functors on the category of finite-dim commutative algebras/k.
Hans-E. Porst <porst@math.uni-bremen.de> pointed me to similar results in his recent paper "On subcategories of the category of Hopf algebras", Arabian Journal of Science and Enginering, (2011), DOI 10.1007/s13369-011-0090-4 Others recommended his article "On corings and comodules", Archivum Mathematicum 42 (2006), no. 4, 419-425. Preprints of these are available at http://www.math.uni-bremen.de/~porst/ I am finding these comments very helpful. Thanks greatly, Terry Bisson [For admin and other information see: http://www.mta.ca/~cat-dist/ ]