Let $Y$ be a complex manifold (possibly non-compact) and $\mathcal{F}_n$, $n=1,2,\dots$ be a directed system of sheaves of $\mathcal{O}_Y$-modules. Suppose that the directed system satisfies the following condition: the direct limit of $\mathcal{F}_n$ in the category of presheaves is a sheaf, i.e., for every open $U\subset Y$ we have \begin{equation} (\varinjlim \mathcal{F}_n)(U) = \varinjlim (\mathcal{F}_n(U)). \end{equation} Is it true that \begin{eqnarray} \label{} H^i(Y,\varinjlim \mathcal{F}_n) = \varinjlim H^i(Y, \mathcal{F}_n), \quad \forall i\geq 0 ? \end{eqnarray} If $Y$ is compact then the answer to my question is yes for any directed system. If we look at the proof (in the stack-project [Tag 01FE]) and try to repeat it for non-compact $Y$, then the argument almost works in a sense that if it was true that the functor of taking global sections commutes with direct limits, then higher cohomologies would also commute with direct limits.

However, in general taking global sections does not commute with direct limits, so unless the directed system has some special property we should not expect the answer to my question to be yes. The condition that I imposed is probably the first thing that one could think of, so probably it is too optimistic. Nevertheless, I was wandering if someone knows a counter example or has a better suggestion.