Surjective geometric morphism ($f^{\ast }\text{ faithful}$)

A geometric morphism is surjective when its inverse-image functor is faithful, equivalently conservative for geometric inverse images. Reflection of isomorphisms of subobjects already proves faithfulness by applying the inverse image to equalizers of parallel arrows. The surjection-embedding factorization of a geometric morphism separates this property from a fully faithful direct image.

For a geometric morphism, the characteristic map of $f^{\ast }\top$ induces a comparison $f^{\ast }{\Omega }_{\mathcal{E}}\to {\Omega }_{\mathcal{F}}$ whose adjoint transpose is $\lambda$. Composing with a characteristic map classifies the inverse-image subobject. Thus monicity of $\lambda$ is equivalent to reflection of equality of subobjects under $f^{\ast }$, and hence to faithfulness of the inverse image: apply reflection to equalizers of parallel arrows. Conversely faithfulness reflects when a pulled-back monomorphism is an isomorphism, by comparing its characteristic map with truth; intersections then reflect equality of arbitrary subobjects.