Note that \(\Fun(C_0,T)\) is a topos. Moreover, \(\Fun^{\kappa}(C,T)\) is closed under colimits and finite limits. It remains to show that \(\Fun^{\kappa}(C,T)\) is accessible. We will do this by cutting it out of \(\Fun(C_0,T)\) by imposing a small set of conditions.Note that a functor \(F \in \Fun^{\lambda}(C,T)\) preserves \(\kappa\)-filtered colimits if and only if \(F\) preserves colimits indexed by \(\kappa\)-filtered posets. Using \(\lambda \gg \kappa\), we may write every \(\kappa\)-filtered poset as a \(\lambda\)-filtered colimit of \(\lambda\)-small \(\kappa\)-filtered colimits, so the condition is equivalent to \(F\) preserving \(\lambda\)-small \(\kappa\)-filtered colimits.There is only a set of isomorphism classes of \(\lambda\)-small \(\kappa\)-filtered posets. For each such poset \(A\), the category \(\Fun(A,C)\) is accessible, so choose a small generating family of diagrams \(\alpha\colon A\to C\). Preservation of the colimit of these generators implies preservation for every \(A\)-diagram, because both sides of the comparison commute with sufficiently filtered colimits of diagrams. We therefore obtain a pullback square where the right vertical functor sends \(F\) to the collection of comparison maps \(\colim_A F\alpha\to F(\colim_A\alpha)\), and the bottom map sends a family of objects to the corresponding family of identity arrows. All three other corners and all functors in this square are accessible. Hence \(\Fun^\kappa(C,T)\) is accessible. Since it is a full subcategory of the topos \(\Fun(C_0,T)\) closed under colimits and finite limits, it is itself a topos.
