Spahn étale topos

Let $H$ be a topos, let $E:=(H/X)_\Box$ ( $\Box$ -closed/étale morphisms over $X$ ).

Lemma

Like every reflective subcategory of a topos $E$ is closed under limits and colimits.

Lemma

In $E$ every monomorphism ist a strong monomorphism?.

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 $H/X$ .