When is the Ind-Completion of a Functor Category Equivalent to the Functor Category of the Ind-Completion?

This paper investigates when the $\kappa$-ind completion of a functor category, denoted $\operatorname{Ind}\kappa(C^I)$, is equivalent to the category of functors from $I$ to the $\kappa$-ind completion of $C$, denoted $\operatorname{Ind}\kappa(C)^I$. While a previous theorem incorrectly claimed this holds for any Cauchy complete category $C$ and $\kappa$-small category $I$, this paper disproves that claim and provides two corrected theorems. The first corrected theorem states the equivalence holds if $C$ has $\kappa$-small colimits and $I$ is $\kappa$-small. The second corrected theorem states the equivalence holds if $C$ is an arbitrary category and $I$ is well-founded and $\kappa$-small. The paper provides counterexamples and discusses the optimality of these conditions.