Geodesic
On coalgebras and type transformations
Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 187-197

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We show that for an arbitrary Set-endofunctor T the generalized membership function given by a sub-cartesian transformation μ from T to the filter functor can be alternatively defined by the collection of subcoalgebras of constant T-coalgebras. Sub-natural transformations ε between any two functors S and T are shown to be sub-cartesian if and only if they respect μ. The class of T-coalgebras whose structure map factors through ε is shown to be a covariety if ε is a natural and sub-cartesian mono-transformation.
Keywords: coalgebra, endofunctor, filter functor, cartesian transformation, crisp 
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Gumm, H. On coalgebras and type transformations. Discussiones Mathematicae. General Algebra and Applications, Tome 27 (2007) no. 2, pp. 187-197. http://geodesic.mathdoc.fr/item/DMGAA_2007_27_2_a2/