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

AbstractThe quasicategory ℚ of all set functors (i.e. endofunctors of the category $\mathbb{SET}$ 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 $\mathbb{F}$ of faithful connected set functors and all natural transformations, and in the quasicategories $\mathbb{B}^{(\kappa)}$ of all set functors and natural transformations which preserve filters of points (up to cardinality κ).