Postnikov completion is topos

The category \(\widehat T\) is presentable because it is a limit of presentable categories along accessible colimit-preserving functors, and its colimits are computed levelwise. We verify descent. For \(X=(X_n)_n\in\widehat T\), slicing commutes with limits of categories and gives Each \(T_{\leq n}\) is an \(n\)-topos, so its slice functor sends colimits to limits. Consequently, if \(X=\colim_iX_i\) in \(\widehat T\), then \begin{align*} \widehat T_{/X} &\iso \lim_n (T_{\leq n})_{/\colim_iX_{i,n}} \\ &\iso \lim_n\lim_i(T_{\leq n})_{/X_{i,n}} \\ &\iso \lim_i\widehat T_{/X_i}. \end{align*} Thus every colimit in \(\widehat T\) is van Kampen, and \(\widehat T\) is a topos by Theorem 2.42.Projection onto the \(n\)-th component identifies \((\widehat T)_{\leq n}\) with \(T_{\leq n}\). Now let \(S\) be Postnikov-complete. A logos morphism into \(S\iso\lim_nS_{\leq n}\) is the same as a compatible collection of left exact colimit-preserving functors into the \(S_{\leq n}\). Moreover, a left exact colimit-preserving functor with \(n\)-truncated target factors uniquely through \(n\)-truncation. We therefore obtain natural equivalences \begin{align*} \Fun_{\Logos}(\widehat T,S) &\iso \lim_n\Fun^{\lex,\colim}((\widehat T)_{\leq n},S_{\leq n}) \\ &\iso \lim_n\Fun^{\lex,\colim}(T_{\leq n},S_{\leq n}) \\ &\iso \Fun_{\Logos}(T,S). \end{align*} Passing to the opposite category of topoi gives the asserted universal property.