  We establish the existence of the universal type structure in presence of
conditioning events without any topological assumption, namely, a type
structure that is terminal, belief-complete, and non-redundant, by performing a
construction \`a la Heifetz & Samet (1998). In doing so, we answer
affirmatively to a longstanding conjecture made by Battigalli & Siniscalchi
(1999) concerning the possibility of performing such a construction with
conditioning events. In particular, we obtain the result by exploiting
arguments from category theory and the theory of coalgebras, thus, making
explicit the mathematical structure underlining all the constructions of large
interactive structures and obtaining the belief-completeness of the structure
as an immediate corollary of known results from these fields.
