Surjective geometric morphism
= Surjective geometric morphism
{title2=$f^*\text{ faithful}$}
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.