This question probably belongs to the very basics of Category Theory, but I have not found an appropriate answer in the latest hours.

Suppose that one has a category $\mathcal{C}$ in which direct limits exist and $\mathcal{I}$ and $\mathcal{K}$ are directed sets. By a *Double Direct System* in $\mathcal{C}$ (w.r.t $\mathcal{I}$ and $\mathcal{K}$), {$A_{i}^{k},f_{ji}^{lk}$} ($i\in \mathcal{I}, k\in \mathcal{K}$), I will mean a collection of objects $A_{i}^{k}\in \mathcal{C} $ and, for each $i\leq j$ and $r\leq s $, morphisms $f_{ji}^{sr}:A_{i}^{r}\rightarrow A_{j}^{s}$ such that:

**1.** $f_{ii}^{rr}=Id:A_{i}^{r}\rightarrow A_{i}^{r}$.

**2.** The composite $A_{i}^{r}\stackrel{f_{ji}^{sr}}{\rightarrow}A_{j}^{s}\stackrel{f_{kj}^{ts}}{\rightarrow}A_{k}^{t}$
equals $f_{ki}^{tr}$.

Note that, fixing $r$, one gets a direct system {$A_{i}^{r}, f_{ji}^{rr}$}, and there is, for each $r\leq s$, an induced map $$f^{sr}:\varinjlim_{i}A_{i}^{r}\rightarrow \varinjlim_{i}A_{i}^{s}$$ which makes {$\varinjlim_{i}A_{i}^{r}, f^{sr}$} a direct system.

Repeating this procedure in the subindices one gets a direct system {$\varinjlim_{r}A_{i}^{r}, f_{ji}$}. My question is: under which conditions (over $\mathcal{C}$, or over the involved morphisms) does one has a (unique?) isomorphism $$\varinjlim_{i}\\ \varinjlim_{r}A_{i}^{r}\cong\varinjlim_{r} \\ \varinjlim_{i}A_{i}^{r}$$ ?

**Remarks:**

**1.** Instead of the given definition one can assume the (apparently?) weaker condition that, fixing subindices $i$, the system {$A_{i}^{r},f_{ii}^{sr}$}$_{r,s\in\mathcal{K}}$ is direct, and analogously for superindices. What do one gets in this case?

**2.** There is a theorem about "Interchange of Limits" in Mac Lane's "Categories for the Working Mathematician", but I do lack of a proper background to quickly see whether this answers my question. Should I do the effort of understanding that result for this?

**3.** (Just for information) The question did arise while I was solving Ex. II.1.10 from Hartshorne's "Algebraic Geometry", as my solution carried me to check the validity of the equation
$$(\varinjlim_{i} \mathcal{F}_{i})_P\cong \varinjlim_i (\mathcal{F}_i)_P$$

where $\mathcal{F}_{i}$ is a direct system of sheaves of abelian groups on a topological space $X$. In this case I could achieve an isomorphism using the universal property for direct limits of abelian groups and using constant sheaves on $X$, but this seems to be quite restrictive.