Showing that inverse limits are unique. Now I am Working on writing a neat answer of this question with the help of the answer mentioned in the link
Proving the universal property for the space $A_{\infty}$ (prototype for the idea of inverse limit).
Here is the next question I want to answer:
Show that if $A^{\infty}$ and $A_{\infty}$ are two different inverse limits for the same tower, then the comparison maps $A_{\infty} \rightarrow A^{\infty}$ and $A^{\infty} \rightarrow A^{\infty}$ are mutually inverse homeomorphisms.  
Could anyone help me in proving this please?
 A: To recap the universal property for easier reference:

We call a space $A_\infty$ and continuous maps $\pi_n: A_\infty \to A_n$ a limit for the inverse system $\left(A_n: n \in \Bbb N; f_{n,m}: A_n \to A_m ( n \ge m) \right)$ when for all $n \ge m$: $f_{n,m} \circ \pi_n = \pi_m$, and for every space $W$ and every family of continuous maps $w_n: W \to A_\infty$ such that also for all $\forall n \ge m: f_{n,m}  \circ w_n = w_m$ we have a unique $\xi: W \to A_\infty$ such that $\forall n: \pi_n \circ \xi = w_n$.

OK, suppose that $A^\infty$ is another inverse limit candidate with projection maps $\pi'_n: A^\infty \to A_n$ (so also obeying the above property, mutatis mutandis), then we have from this property applied to $W=A^\infty, w_n= \pi'_n$, a continuous $\xi: A^\infty \to A_\infty$ that obeys 
$$\forall n: \pi_n \circ \xi = \pi'_n$$
Similarly the universal property for $A^\infty$ and $W=A_\infty, w_n=\pi_n$ gives us a (other name to avoid confusion) $\psi: A_\infty \to A^\infty$ such that
$$\forall n: \pi'_n \circ \psi = \pi_n$$
Note then that for all $n$, $\psi \circ \xi$ obeys
$$\pi'_n \circ (\psi \circ \xi)= (\pi'_n \circ \psi) \circ \xi = \pi_n \circ \xi = \pi'_n = \pi'_n \circ \text{id}_{A^\infty}$$
so $\psi \circ \xi$ fulfills the universal property commutative relations for the map from $A^\infty$ to itself when we use the identity maps on all $A_n$ as input (so with $W=A^\infty, w_n = \text{id}_{A_n}$), but $\text{id}_{A^\infty}$ also does, so the unicity clause for that "identity situation" gives us that $$\psi \circ \xi = \text{id}_{A^\infty}$$
and reversing the order of $\xi$ and $\phi$ and applying the same argument, mutatis mutandis, we also get
$$\xi \circ \psi = \text{id}_{A_\infty}$$
and the connecting maps are each others's inverse so $A^\infty \simeq A_\infty$.
This is how basically all proofs of essential unicity of objects defined by universal properties go (in category theory). This exercise is a taste of that.
