Spahn reflective subcategory of a topos (Rev #1, changes)

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A reflective subcategory of a topos is a topos if the reflector is left exact.

Let $H\simeq Sh_J(H)$ be a topos for the canonical topology $J$. Let

be the inclusion of a reflective subcategory into a topos. Let

the sheafification functor.

$H$ and $E$ are total. Since $H$ is a topos, $H$ is total, since $E$ is a reflective subcategory of a total category $E$ is total. This implies the the Yoneda embeddings of $E$ and $H$ both posess left adjoints

apparently $a_H=Sh_J(H)$.

Let $R_!:=Lan_{Y_E} Y_H R$ then we have $L\circ a_H\circ R_!$ is the identity on representables and

is a natural equivalence.