Weakly Terminal Objects in Quasicategories of $\mathbb{SET}$ Endofunctors

The quasicategory ℚ of all set functors (i.e. endofunctors of the category of all sets and mappings) and all natural transformations has a terminal object – the constant functor C₁. We construct here the terminal (or at least the smallest weakly terminal object, which is rigid) in some important subquasicategories of ℚ – in the quasicategory of faithful connected set functors and all natural transformations, and in the quasicategories of all set functors and natural transformations which preserve filters of points (up to cardinality κ).Mathematics Subject Classifications (2000) 18A22, 18A25.Libor Barto: This work was completed with the support of the Grant Agency of the Czech Republic under the grant 201/02/0148; supported also by MSM 113200007.