Corollary 2.17.
Let \(T\) be a topos and let \(L\colon T \to T'\) be a left exact functor localization. Then also \(T'\) is a topos.
Proof
Let \(R\colon T' \hookrightarrow T\) denote the fully faithful right adjoint to \(L\). Then this adjunction induces an adjunction
\[L_*\colon \Ar(T) \rightleftarrows \Ar(T')\colon R_*\]
on arrow categories. Moreover, since both \(L\) and \(R\) preserve pullback squares, this adjunction restricts to an adjunction \[\Ar^{\pb}(T) \rightleftarrows \Ar^{\pb}(T').\]
The claim now follows from Proposition 2.16, as \(\Ar^{\pb}(T')\) inherits its colimits from \(\Ar^{\pb}(T)\) and these are compatible with the ones in \(\Ar(T')\).