Is there a special name for a category with all morphisms being morphisms from an initial object?

Question

Let $\mathcal C$ be a (small) category whose only non-identity morphisms are morphisms from an initial object.

I guess $\mathcal C$ can be obtained by adjoining an initial object to a discrete category.

Is there a special name for this type of category? (I might have called it a "bouquet of morphisms", but this doesn't capture the orientation of the morphisms.)

Answer 1

You could call it the "walking cone." See here.

Answer 2

The result is necessarily a partial order. In domain theory, this is called a flat domain.