nLab
strict initial object

Definition

An initial object $\varnothing$ is called a strict initial object if any morphism $x\to \varnothing$ must be an isomorphism.

Examples

The initial objects of a poset, of Set, Cat, Top, and of any topos, more generally of any extensive category) are strict.

At the other extreme, a zero object is only a strict initial object if the category is trivial (equivalent to the terminal category).