Assume there are two directed systems $(A\_i,f\_{ij})$ and $(A'\_i,f'\_{ij})$ of groups over the directed set $(\mathbb{N},\le)$, such that for any $i,j$ there exists isomorphisms $\phi\_{ij}:A\_i\to A\_i'$ and $\psi\_{ij}:A\_j\to A'\_j$ such that $f'\_{ij}\circ\phi\_{ij}=\psi\_{ij}\circ f\_{ij}$. Is it true that $\varinjlim A\_i$ is isomorphic to $\varinjlim A'\_i$?


If it is true, is there some preferred isomorphism? What about direct limits in other categories and other directed sets?



Compare direct limits of two pairwise isomorphic direct systems
This is false.


**Example.** Let $A\_i = A'\_i = \mathbf Z^{(\mathbf N)}$ be an infinitely generated free abelian group. For $i < j$, let $f\_{ij} \colon A\_i \to A\_j$ be the map that kills the first $j$ coordinates, and $f'\_{ij} \colon A'\_i \to A'\_j$ the map that kills the even coordinates up to $2j$:
\begin{align}
f\_i((a\_j)\_j) &= (0,\ldots,0,a\_{j+1},\ldots),\\
f'\_i((a\_j)\_j) &= (a\_1,0,a\_3,0,\ldots,a\_{2j-1},0,a\_{2j+1},a\_{2j+2},\ldots).
\end{align}
It is clear that the $A\_i$ and $A'\_i$ are pairwise isomorphic (even in finite subsets). However, the colimits are not isomorphic:
$$A = \underset{\longrightarrow}{\operatorname{colim}} A\_i = 0,$$
since an element $(a\_1,\ldots,a\_j,0,0,\ldots)$ in some $A\_i$ disappears in $A\_j$. On the other hand,
$$A' = \underset{\longrightarrow}{\operatorname{colim}} A'\_i = \mathbf Z^{(2\mathbf N-1)}$$
consists of all finite sequences $(a\_1,\ldots) \in \mathbf Z^{(\mathbf N)}$ supported on odd coordinates only. Indeed, the even coordinates all die by the same argument, and the odd ones never change in the colimit.


