let $X$ be small-complete and $J$ small categories, then we can define the functor $$ \operatorname{Lim} \colon X^J \to X $$ which sends each functor $$ F \colon J \to X $$ to its limit.
I have been thinking, when does the $\operatorname{Lim}$ functor have a limit? If $X^J$ is small, by completeness of $X$ there must exist the limit of $\operatorname{Lim}$, and then my question is how to find this limit?