Example 4.8.
Let \(T = \PSh(C)[\Sigma^{-1}]\) be a left exact localization of a presheaf category. Then the inclusion
\[\Pt(T) \quad \simeq \quad \Fun_{\bbLog}(T,\An) \quad \hookrightarrow \quad \Fun_{\bbLog}(\PSh(C),\An) \quad \simeq \quad \Pro(C)\catop\]
identifies \(\Pt(T)\) with the \(\Sigma\)-local pro-objects in \(C\), i.e. those formal systems \((X_i)_{i \in I}\) for which the functor
\[\PSh(C) \to \An, \qquad F \longmapsto \colim_i F(X_i)\]
inverts all morphisms in \(\Sigma\). Here we use the standard correspondence between pro-objects of \(C\) and left exact colimit-preserving functors \(\PSh(C)\to\An\): a pro-object \((X_i)_i\) determines the displayed functor, and every such functor arises uniquely in this way.