Spahn subtopos (changes)

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Theorem (Elephant A.3.4.9, p.192)

Let $L\stackrel{i}{\hookrightarrow}E$ be a reflective subcategory of a topos such that the reflector $l$ is cartesian.

Then $L$ is a topos and $l$ preserves finite limits (i.e. $(l\dashv i)$ is a geometric morphism).

Theorem (Elephant A.4.3.9, p.192)

Let $(l\dashv i):L\stackrel{i}{\hookrightarrow}E$ be a reflective subcategory of a topos such that the monad $i\circ l$ is cartesian (i.e. $i\circ l$ preserves pullbacks).

Then $L$ is a topos and $l$ preserves finite limits (i.e. $(l\dashv i)$ is a geometric morphism).

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