The following lemma improves on the statement * *A reflective subcategory of a topos is a topos if the reflector is left exact.* +-- {: .num_lemma} ###### Lemma Let $(L\dashv R):E\stackrel{\R}{\hookrightarrow} H$ be a reflective subcategory of a topos. Then $E$ is a topos if $L$ preserves pullbacks in the image of $a_H\circ R_!$ where * $(a_H\dashv Y_H):H\to Psh(H)$ is the left adjoint of the Yoneda embedding of $H$. * $R_!:=Lan_{Y_E} Y_H\circ R$ is the left Kan extension of $Y_H\circ R$ along the Yoneda embedding of $E$. $$\array{ Psh(E)&\stackrel{a_E\dashv Y_E}{\to}&E \\ \downarrow^{R_!}&&\downarrow^{L\dashv R} \\ Psh(H)&\stackrel{a_H\dashv Y_H}{\to}&H }$$ =-- +-- {: .proof} ###### Proof 1. The Yoneda embeddings of $E$ and $H$ both posess left adjoints: $H$ and $E$ are [[total category|total]]: Since $H$ is a topos, $H$ is total, since $E$ is a reflective subcategory of a total category $E$ is total. By the adjoint functor theorem for total categories this implies that the Yoneda embeddings of $E$ and $H$ both posess left adjoints. 1. We have $a_E\simeq L\circ a_H\circ R_!$. If this composite is left exact it exhibits $E$ as a left exact localization of a category of presheaves and hence in this case $E$ is a topos. 1. $a_H\circ R_!$ sends colimits to limits, since $R_!$ (as every Yoneda extension) commutes with colimits and $a_H$ as a left adjoint sends colimits to limits. 1. Hence $a_E\simeq L\circ a_H\circ R_!$ is left exact iff $L$ preserves limits in the image of $a_H\circ R_!$. 1. Since a reflector always preserves terminal objects (and all finite limits can be constructed from pullbacks and the terminal object), it is sufficient to check if $L$ preserves pullbacks in the image of $a_H\circ R_!$. =-- (If $\circ$ denotes a duality and $l\dashv r$ then $r^\circ\dashv l^\circ$. Hence if $R_!$ has a left adjoint, then $R$ has a right adjoint. Every subcategory of a category of presheaves which is reflective and coreflective is itself a category of presheaves (this is quoted at [[nLab:reflective subcategory]] as Bashir Velebil). In particular $E$ is a topos.)