Postnikov complete and hypercomplete topoi

We denote by \(\Topos^{\post} \subseteq \Topos^{\hyp} \subseteq \Topos\) the full subcategories of Postnikov-complete and hypercomplete topoi. These two inclusions have right adjoints sending \(T\) to \(T^{\hyp} = T_{\leq \infty}\) and \(T \mapsto \widehat{T}\).