Given a finitely complete category $C$, is there a way to obtain a "delexing" $\operatorname{Delex}(C)$ of $C$ so that the following property holds; $\operatorname{Lex}(C,D) \cong [\operatorname{Delex}(C),D]$ for all finitely-complete (for simplicity) $D$? Where on the left we have the category of left exact functors and natural transformations between $C$ and $D$ and on the right the usual functor category.

The example I have in mind here is when $C$ is the finitely complete category obtained from the finite limit theory of categories. If it's not possible in general, is there a subset of finitely complete categories for which it is possible?