Remark 6.140.
The category \(\Fun^{\kappa}(C,T)\) has a universal property: for every topos \(S\), there is a natural equivalence
\[\Geom(\Fun^{\kappa}(C,T), S) \simeq \Fun^{\kappa}(C, \Geom(T,S)).\]
Both sides are naturally a full subcategory of \(\Fun(C \times S, T)\), and one can check that in both cases we precisely get those functors \(C \times S \to T\) that preserve \(\kappa\)-filtered colimits in the left variable, and preserve finite limits and colimits in the second variable.