The notion of an endofunctor having "greatest subcoalgebras" is introduced as a form of comprehension. This notion is shown to be instrumental in giving a systematic and abstract proof of the existence of limits for coalgebras -- proved earlier by Worrell and by Gumm & Schroder. These insights, in dual form, are used to reinvestigate colimits for algebras in terms of "least quotient algebras" -- leading to a uniform approach to limits of coalgebras and colimits of algebras. Finally, at an abstract level of fibrations, an equivalence is established between having greatest subcoalgebras (in a base category of types) and greatest invariants (in a total category of predicates)
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