In "Topoi: The Categorial Analysis of Logic" by R. Goldblatt the $false: 1 \to \Omega$ truth value is defined as the characteristic arrow of the arrow $0_1: 0 \to 1$. This definition requires that the arrow $0_1$ is mono. Why is this true?
UPD:
For the proof see theorem 6.3 in "Elementary categories, elementary Toposes" by C. McLarty. So thank you Zhen for referring to the notion of strict initial object.
In a topos (or more generally, any cartesian closed category with an initial object), $0$ is a strict initial object. That is, every morphism $X \to 0$ is an isomorphism. In particular, any parallel pair $X \rightrightarrows 0$ is degenerate. It follows that every morphism $0 \to Y$ is a monomorphism.
Let $\mathcal{C}$ be a cartesian closed category, let $0$ be an initial object in $\mathcal{C}$, and let $f : X \to 0$ be a morphism in $\mathcal{C}$. Then we have $\langle \mathrm{id}_X, f \rangle : X \to X \times 0$ and a projection $p : X \times 0 \to X$. Of course, $p \circ \langle \mathrm{id}_X, f \rangle = \mathrm{id}_X$. But $X \times 0 \cong 0$ (because $\mathcal{C}$ is cartesian closed), so $X \cong 0$ as well. Hence $f : X \to 0$ is an isomorphism.