About direct limit of groups

Question

Let $G_i$ be sequence of groups for $i\in \mathbb N$ and Let $\phi_i$ be a monomorphism from $G_i$ to $G_{i+1}$.

Let $\Sigma$ be the direcet limits of $G_i$ under the embeddings of $\phi_i$.

Let $\varphi_i$ be another monomorphism from $G_i$ to $G_{i+1}$ s.t.

$$\phi_i(G_i)=\varphi_i(G_i)$$

i.e their image are same. (but not element wise).

If $\Sigma '$ is the direct limits of $G_i$ under the embeding of $\varphi_i$ then can we say that $\Sigma\cong \Sigma '$ ?

Answer 1

Another simple example (abelian torsion free of rank 2).

Let $G_i=\mathbf{Z}^2$, $\phi_i(x,y)=\phi(x,y)=(px,y)$ and $\varphi_i(x,y)=\varphi(x,y)=(py,x)$. Then $\Sigma_\phi=\mathbf{Z}[1/p]\times\mathbf{Z}$ while $\Sigma_\varphi=\mathbf{Z}[1/p]^2$ (note that $\varphi^2(x,y)=(px,py)$).

[Let us observe that the condition that $\phi_i$ and $\varphi_i$ have the same image is somewhat unstable, since it does not imply that $\phi_{i+1}\circ\phi_i$ and $\varphi_{i+1}\circ\varphi_i$ have the same image.]