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 or even wikipedia's talk page for Čech Cohomology, 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.

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 or even wikipedia's talk page for Čech cohomology, 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.