Well-founded coalgebras generalize well-foundedness for graphs, and they capture the induction principle for well-founded orders on an abstract level. Taylor’s General Recursion Theorem shows that, under hypotheses, every well-founded coalgebra is parametrically recursive. We give a new proof of this result, and we show that it holds for all set functors, and for all endofunctors preserving monomorphisms on a complete and well-powered category with smooth monomorphisms. The converse of the theorem holds for set functors preserving inverse images. We provide an iterative construction of the well-founded part of a given coalgebra: It is carried by the least fixed point of Jacobs’ next-time operator.