I need a term for a concrete category over Set with a distinguished object $i,$ such that the morphisms from $i$ correspond to elements of the other objects. More formally, for every object $o$ and every element $x \in o,$ there is exactly one morphism $m_{o,x}: i \to O.$ Is this an established concept, and, if so, what is the nomenclature.
Note that the terms initial and pointed already have other meanings in category theory.
The question arose while revising https://arxiv.org/abs/1801.05775
If what you're after is a concrete category where $U$ is representable, then such categories are called representably concrete.