Let $X$ be a Noetherian scheme over a Noetherian ring $R$ and $(F_{\alpha})_{\alpha \in I}$ a direct system of $O_X$-module sheaves on $X$. I'm looking for source literature where I can find a proof of the fact that colimits of sheaves commute with sheaf cohomology, ie that for all $i\ge0$ the canonical morphism of $R$-modules

$$\varinjlim_i H^{i}(X, F_{\alpha}) \to H^i(X, \varinjlim_{\alpha} F_{\alpha}) $$

induced by $F_{\alpha} \to \varinjlim_{\alpha} F_{\alpha}$, applying the naturality of cohomology functor and the universal property of colimits. That's of course a well known fact used by people involved in research on algebraic geometry literally by reflex but I nowhere found a well explained proof of this isomorphism.