Zero object is the only exponentiable object in a pointed category
= Zero object is the only exponentiable object in a pointed category
If the <terminal object> is also initial and $E$ is exponentiable, the left adjoint $-\times E$ preserves the initial object. But $0\times E\cong E$ because $0$ is terminal, while preservation of initiality gives $0\times E\cong0$. Hence $E\cong0$.