Suppose we have an inverse system of compact Hausdorff spaces $\lbrace X_i , \varphi_{ij} \rbrace_{i\in I}$ and that each space has a presheaf $\Gamma_i$  assigned to it in such a way that $\Gamma_i(\varphi_{ij}(U))=\Gamma_j(U)$ whenever $i\leq j$. Then $X:=\varprojlim X_i$ has a presheaf $\Gamma$ defined on it by $\Gamma(U):=\Gamma_i(\varphi_i(U))$ where $\varphi_i:X\to X_i$ is the map from $X$ as an inverse limit; this is well-defined since $\varphi_{ij}(\varphi_j(U))=\varphi_i(U)$ whenever $i\leq j$, which makes $\Gamma_i(\varphi_i(U))=\Gamma_i(\varphi_{ij}(\varphi_j(U)))=\Gamma_j(\varphi_j(U))$.

In this situation, does Čech cohomology satisfy a continuity property? That is, is it true that $\breve{H}^*(X,\Gamma)=\varinjlim \breve{H}^*(X_i,\Gamma_i)$? I've seen it claimed in some places, such as [this question][1] or even wikipedia's [talk page for Čech cohomology][2], that Čech cohomology satisfies some kind of continuity property for "nice enough" spaces, but I can't seem to find any clear reference for this fact. The paper that question refers to seems to be concerned with a more general situation involving triangulable pairs, and I can't fully make sense of it.


  [1]: https://mathoverflow.net/questions/257588/continuity-of-alexander-spanier-cohomology
  [2]: https://en.wikipedia.org/wiki/Talk%3A%C4%8Cech_cohomology#Inverse_Limits