  For a strongly connected category $\mathcal C$ with pair-wise coproducts, we
introduce a cosimplicial object, which serves as a sort of resolution for
computing higher derived functors of ${\sf lim} : \mathrm{Ab}^{\mathcal C}\to
\mathrm{Ab}$. Applications involve K\"unneth theorem for higher limits and
${\sf lim}$-finiteness of ${\bf fr}$-codes. A dictionary for the ${\bf
fr}$-codes with words of length $\leq 3$ is given.
