Proposition 6.145.
Let \(C\) be compactly assembled, and let \(D\) be presentable. Then there is an equivalence
\[\FunL(\Fun^{\omega}(C,\An),D) \simeq \Fun_{\omega}(C\catop,D),\]
where the RHS means those functors that preserve cofiltered limits.
Proof sketch
If \(C = \Ind(C_0)\) is compactly generated, we have \(\Fun^{\omega}(C,\An) = \Fun(C_0,\An) = \PSh(C_0\catop)\), whereas \(\Fun_{\omega}(C\catop,D) = \Fun(C_0\catop,D)\). We would like to say: the claim now follows as every \(C\) is a retract of a compactly generated one. However, for this reasoning to work, we need to find a way to write down the comparison map for the compactly generated case in a way that does not rely on picking an identification \(C = \Ind(C_0)\).To this end, consider the category
\[\Fun^+(C,\An) := \Fun^{\omega}(C,\An) \cup \Fun^{\rep}(C,\An) \subseteq \Fun(C,\An).\]
Consider a functor \(F\colon \Fun^+(C,\An) \to D\). Then, if \(C = \Ind(C_0)\), one can show that: - \(F\) is left Kan extended from \(\Fun^{\rep}\) if and only if \(F\vert_{\Fun^{\omega}}\) preserves colimits.
- \(F\) is right Kan extended from \(\Fun^{\omega}\) if and only if \(F\vert_{\Fun^{\rep}}\) preserves cofiltered limits.
- \(F\) is left Kan extended from \(\Fun^{\rep}\) and \(F\vert_{\Fun^{\rep}}\) preserves cofiltered limits.
- \(F\) is right Kan extended from \(\Fun^{\omega}\) and \(F\vert_{\Fun^{\omega}}\) preserves colimits.