  In this article, we combine Sweedler's classic theory of measuring coalgebras
-- by which $k$-algebras are enriched in $k$-coalgebras for $k$ a field -- with
the theory of W-types -- by which the categorical semantics of inductive data
types in functional programming languages are understood. In our main theorem,
we find that under some hypotheses, algebras of an endofunctor are enriched in
coalgebras of the same endofunctor, and we find polynomial endofunctors provide
many interesting examples of this phenomenon. We then generalize the notion of
initial algebra of an endofunctor using this enrichment, thus generalizing the
notion of W-type. This article is an extended version of arXiv:2303.16793, it
adds expository introductions to the original theories of measuring coalgebras
and W-types along with some improvements to the main theory and many explicitly
worked examples.
