 We write 0 for ∅. Each of the maps shown is an inclusion, by standardness. Let F* be the union of the increasing sequence  Fn0 of sets. Then F(F*) = F*   by continuity.    So  (F*, id)  is an algebra for F. To check initiality, let  (A, a) be an algebra for F,  so  a : Fa → a. Define maps  gn : Fn(0)   → A   by recursion, with    g0 : 0   → A   the empty function (this is what initiality of ∅ amounts to), and    gn+1 =    a ⋅ Fgn. Check that we have an increasing sequence of functions 