Let $H$ be a topos, let $E:=(H/X)_\Box$ ($\Box$-closed/étale morphisms over $X$). +-- {: .num_lemma} ###### Lemma Like every [[reflective subcategory of a topos]] $E$ is closed under limits and colimits. =-- +-- {: .num_lemma} ###### Lemma In $E$ every monomorphism ist a [[strong monomorphism]]. =-- +-- {: .num_proof} ###### Proof $E$ is a topos and hence any monomorphism in $H$ is strong. Let $$\array{ Q&\to&Y \\ \downarrow&\nearrow&\downarrow \\ Z&\to&W }$$ be a solved lifting problem with $Y\to W$ an etale monomorphism, $Z\to W$ an etale morphism, and $Q\to Z$ an epimorphism. Then by the left cancellation property also $Z\to Y$ is etale. This remains true if we consider the lifting problem in $H/X$. =--