The boolean domain object $\mathbf{2}$ is the disjoint coproduct of the terminal object $1$ with itself.

… (One should be able to define binary coproducts using the dependent sum functor and the boolean domain object, as dependent sums exist in topoi and cartesian closed categories.)

Boolean domain objects are the categorical semantics of the boolean domain in type theory. The inductive property of the boolean domain type, case analysis or if/else expressions, corresponds to the initiality of the boolean domain object in the subcategory of triples $(A, t:1\rightarrow A, f:1\rightarrow A)$ representing bi-pointed objects, similar to how the principle of induction over natural numbers corresponds to the initiality of the natural numbers object in the subcategory of triples $(A, q:1\rightarrow A, f:A\rightarrow A)$ representing infinite sequences.