Coalgebras, Counits, and the Fundamental Coalgebra Theorem — Examples

Compute group-like and primitive elements in finite-dimensional small coalgebras, quotient a comatrix coalgebra by a counital coideal, and show why ε(I)=0 is indispensable. The fundamental theorem is over fields, not arbitrary rings.

These are planned worked examples and counterexamples for coalgebras-counits-and-the-fundamental-coalgebra-theorem, with empty item lists pending normal authoring and proof review. Each example must calculate from the stated hypotheses, and each counterexample must verify both its premises and the failed conclusion. This companion is a dependency leaf and supplies no theorem to another page.