In the book Tensor Categories, Exercise 1.9.4 is to prove the fundamental theorem of coalgebras. They give the following hint:
This hint doesn't make sense to me. You can always add and subtract a term from the sum to make the span larger, so this supposed subcoalgebra depends on how I represent the comultiplication.
In particular, in order for the span to always give a subcoalgebra, it would need to be the case that every finite dimensional subspace containing the subcoalgebra generated by $c$ is itself a subcoalgebra, which is clearly false.
Am I missing something or misunderstanding the hint? Do they mean the intersection of all such spans, or something like that?
I know how to prove the exercise without the hint, such as how its proven in Montgomery's book, but this hint is very weird to me and isn't mentioned in the errata.