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Suppose $T$ is a (grothendieck) topos, and suppose $C\subseteq T$ is a full subcategory. If the inclusion $C\hookrightarrow T$ has a lex left adjoint, is $C$ necessarily a topos?
What if the embedding is not necessarily full?
As you know, Grothendieck toposes are the exact reflective subcategories of presheaf toposes. So if $T$ is a Grothendieck topos, there exists a small category $B$ and a left exact reflective inclusion $T \hookrightarrow PSh(B)$. Since left exact reflective inclusions are closed under composition, it follows that also
$C \hookrightarrow T \hookrightarrow PSh(B)$is a left exact reflective inclusion into a presheaf topos.
Thanks!
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