The two-valued 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 two-valued object, as dependent sums exist in topoi and cartesian closed categories.)

Two-valued objects are the categorical semantics of the two-valued type in type theory. The inductive property of the two-valued type, case analysis or if/else expressions, corresponds to the initiality of the two-valued 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.