Please forgive me if this is not the right forum for this question.

Let $$ X = \cdots \rightarrow X_n \rightarrow X_{n-1} \rightarrow \cdots \rightarrow X_0 = \ast$$ be a tower of fibrations of simplicial sets. I am trying to understand a proof of Milnor's exact sequence found in Goerss-Jardine, but I am having some trouble and have been thinking about it for quite some time.

In the proof one considers the following model for the homotopy inverse limit of X, namely as the pullback of two morphisms where the first is $$j: \Pi_n Hom(\Delta^1,X_n) \rightarrow \Pi_n X_n \times X_n$$ given by the inclusion $$\partial \Delta^1 \rightarrow \Delta^1$$ and the second is given by $$(1,q): \Pi_n X_n \rightarrow \Pi_n X_n \times X_n .$$ Here , $(1,q)$ is the product of the maps $$(1,p_n): \Pi_n X_n \rightarrow X_n \times X_{n+1} \rightarrow X_n \times X_n , $$ where the first map is projection followed by the identity times $p_n$. $p_n$ is the map $X_{n+1} \rightarrow X_n$ coming from the tower of fibrations. Let us call this pullback for $T(X).$ Then it is easy to see that the map $$T(X) \rightarrow \Pi_n X_n $$ is a fibration with fiber $$\Pi_n \Omega X_n .$$ Then, in the long exact sequeence in homotopy induced by this fibration, it is claimed that the boundary map $$\partial_i: \pi_{i+1}(\Pi_n X_n) \rightarrow \pi_i (\Pi_n \Omega X_n) $$ can be identified, under the iso $\pi_i(\Omega X) \cong \pi_{i+1}(X)$' with the map $$f: \pi_{i+1} (\Pi_n X_n) \rightarrow \pi_{i+1}(\Pi_n X_n)$$ where $$f((a_i))= (a_i -p_n(a_{i+1})) ,$$ i.e a map whose kernel is the limit of the tower pf abelian groups $\pi_{i+1}(X). $

My question is, how can the boundary map be identified with the map $f$ as above? I tried to use naturality but to no real luck.

I would be very grateful for help.