Every object isomorphic to a coproduct of terminal objects

Question

Let $C$ be a category with coproducts and terminal object $1$ such that every object in $C$ is isomorphic to a coproduct of $1$ with itself (indexed over some set). Is there a special name for such a category?

EDIT: Interested specifically in the case in which $C$ is a topos.

Answer 1

These are precisely the cocartesian categories $\mathbf{C}$ for which the (singleton set containing the) terminal object $\{ 1 \}$ is a colimit-dense generator. See, for instance, §III.7 of Adámek–Tholen's Total Categories with generators. I do not believe there is a more specific name; examples seem sparse.