  We present a construction of W-types in the setoid model of extensional
Martin-L\"of type theory using dependent W-types in the underlying intensional
theory. More precisely, we prove that the internal category of setoids has
initial algebras for polynomial endofunctors. In particular, we characterise
the setoid of algebra morphisms from the initial algebra to a given algebra as
a setoid on a dependent W-type. We conclude by discussing the case of free
setoids. We work in a fully intensional theory and, in fact, we assume identity
types only when discussing free setoids. By using dependent W-types we can also
avoid elimination into a type universe. The results have been verified in Coq
and a formalisation is available on the author's GitHub page.
