In Hartshorne's book, it states that there is a natural transformation between two $\delta-$functors $\text{lim}_{\rightarrow} H^{i}(X, .)$ and $H^{i}(X,\text{lim}_{\rightarrow} .)$. Here $\text{lim}_{\rightarrow}$ denotes the direct limit of a direct system, and $H^{i}(X, .)$ means the cohomology of sheaves on a topological space $X$, so both functors have type $\mathfrak{ind}_A(\mathfrak{Ab}(X)) \to \mathfrak{Ab}$.
Since I am not quite familiar with the language of category, I can't see this relationship clearly. Hope someone could help; thanks!
The point is that direct limits (that are actually a special case of colimits) have a universal property : a map out of the direct limit is entirely determined by what it does on the elements of the system, and any coherent system of maps out of the system determines a map out of the direct limit. If you know this, you can keep on reading, if you don't, you should read up on direct limits first to see the precise statement
Herr if you have a directed system of sheaves $\{\mathcal{F}_i\}$ then the inclusions $\mathcal{F}_i \to \varinjlim_j \mathcal{F}_j $ form a coherent system of maps, therefore by functoriality, so do the induced maps $H^k(X, \mathcal{F}_i)\to H^k(X, \varinjlim_j\mathcal{F}_j)$, so by the universal property they induce a single map $\varinjlim_i H^k(X, \mathcal{F}_i)\to H^k(X, \varinjlim_j \mathcal{F}_j)$.
At this point, since everything was defined using universal properties, it will be very easy that this is indeed a natural transformation. That it is a natural transformation of $\delta$-functors, i.e. that it respects the connecting morphisms should follow from the definition of the connecting morphisms for $\varinjlim H^k(X, -)$ and from the universal properties too.