Proposition 4.18.
For every logos \(S\), restriction along \(T \to T/K\) induces a fully faithful functor
\[\Fun_{\bbLog}(T/K, S) \quad \hookrightarrow \quad \Fun_{\bbLog}(T,S).\]
Its essential image consists of those morphisms of logoi \(\phi\colon T \to S\) that invert \(K\) (i.e. \(K \subseteq \ker(\phi)\)).
Proof
By definition of localization, restriction along \(T \to T/K\) induces a fully faithful functor
\[\Fun(T/K,S) \hookrightarrow \Fun(T,S),\]
with essential image those functors \(T \to S\) that invert \(K\). It remains to show that a functor \(T/K \to S\) preserves colimits and finite limits if and only if the composite \(T \to T/K \to S\) does.The `only if' direction is clear. For the `if' direction, note that every diagram in \(T/K\) may be regarded as a diagram in \(T\) via the inclusion. Since \(T \to T/K\) preserves colimits and finite limits, colimits and finite limits in \(T/K\) may be computed by forming them first in \(T\) and then localizing. Thus if \(T \to S\) is a logos morphism, then so is \(T/K \to S\).