  We describe the Lelek fan, a smooth fan whose set of end-points is dense, and
the Poulsen simplex, a Choquet simplex whose set of extreme points is dense, as
Fra\"{i}ss\'e limits in certain natural categories of embeddings and
projections. As an application we give a short proof of their uniqueness,
universality, and almost homogeneity. We further show that for every two
countable dense subsets of end-points of the Lelek fan there exists an
auto-homeomorphism of the fan mapping one set onto the other. This improves a
result of Kawamura, Oversteegen, and Tymchatyn from 1996.
