I'm running into a little trouble with the proof of Proposition 11.1 on page 64 of the first chapter of Goerss and Jardine's book Simplicial Homotopy Theory.
Proposition 11.1. Suppose $X$ is a Kan complex, then the unit $$\eta_X:X\to S|X|$$ arising from the adjunction $|-|:\mathrm{sSet}\leftrightarrows\mathrm{Top}:S$ is a weak equivalence of simplicial sets, in the sense that it induces isomorphisms in all possible homotopy groups.
The proof is by induction. One shows that the path components are in one to one correspondence, i.e. one proves the statement for the $0$-th homotopy groups, and one hopes to apply the long exact homotopy sequence of the path fibration to get to higher homotopy groups by shifting the dimension using the loop space. However, they draw a diagram with one map I don't know how to interpret properly: $$\begin{matrix} \pi_{n+1}(X,x) & \xrightarrow{(\eta_X)_*} & \pi_{n+1}(S|X|,x) \\ \partial\downarrow && \downarrow{\color{red}\partial}\\ \pi_{n}(\Omega X,x) & \xrightarrow{(\eta_{\Omega_x X})_*} & \pi_{n}(S|\Omega_x X|,x) \end{matrix}$$ (I have written abusively $x$ everywhere, when it should be something like $\eta_X(x)$, or the constant loop at $x$) The problem I have is with the red ${\color{red}\partial}$. In the paragraph below I explain how I believe this diagram arises. However, once this is established I still don't see how to finish:
We want to perform a dimension shift argument using the long exact sequence of the path fibration of $S|X|$, but this would require we replace the red map $$\pi_{n+1}(S|X|,x)\xrightarrow{{\color{red}\partial}}\pi_{n}(S|\Omega_x X|,x)$$ with the connecting homomorphism arising from the path fibration of $S|X|$, i.e. with $$\pi_{n+1}(S|X|,x)\xrightarrow{{\color{blue}\partial}}\pi_{n}(\Omega_xS| X|,x)$$ it would have to land in $\pi_{n}(\Omega_x S| X|,x)$, but then the arrow $(\eta_{\Omega_X})_*$ would have the wrong target.
I doubt that there should be an isomorphism of simplicial sets $\Omega_x S|X|\simeq S|\Omega_x X|$. What follows is presumably how they got the diagram.
Theorem 1.10 tells us that the geometric realization $|P_xX|\to |X|$ of the path space fibration $P_xX\to X$ is a Serre fibration, and it easily follows that $S|P_xX|\to S|X|$ is a Kan fibration. The loop space at $x$ is defined by pullback (a finite limit): $$\begin{matrix} \Omega_x X & \rightarrow & P_xX\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & X \end{matrix}$$ Also, since $|-|$ preserves finite limits by Proposition 2.4, we get a pullback diagram upon applying geomtric realization to the previous diagram: $$\begin{matrix} |\Omega_x X| & \rightarrow & |P_xX|\\ \downarrow & &\downarrow\\ \lbrace\mathrm{pt}\rbrace &\rightarrow & |X| \end{matrix}$$ $S$ is a right adjoint, hence preserves limits, and we get a further pullbakc diagram $$\begin{matrix} S|\Omega_x X| & \rightarrow & S|P_xX|\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & S|X| \end{matrix}$$ As was noted earlier, the vertical map on the right is a fibration, and all simplicial sets involved are Kan complexes, so this gives us the natural setting for the long exact sequence of homotopy groups of a fibration. This diagram has a natural map coming from the original diagram $$\begin{matrix} \Omega_x X & \rightarrow & P_xX\\ \downarrow & &\downarrow\\ \Delta^0 &\rightarrow & X \end{matrix}$$ induced by the unit $\eta$ of the adjunction, and the naturality of the long exact sequence of a fibration gives a ladder diagram $$ \begin{matrix} \cdots\rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & \pi_{n}(P_xX,x) & \rightarrow & \pi_{n}(X,x) & \xrightarrow{\partial} & \pi_{n-1}(\Omega_xX,x) &\rightarrow\cdots \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ \cdots\rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{{\color{red}\partial}} & \pi_{n}(S|\Omega_xX|,x) & \rightarrow & \pi_{n}(S|P_xX|,x) & \rightarrow & \pi_{n}(S|X|,x) & \xrightarrow{{\color{red}\partial}} & \pi_{n-1}(S|\Omega_xX|,x) &\rightarrow\cdots \end{matrix}$$
This at least clears up what the original red ${\color{red}\partial}$ meant, and how the diagram was conceived.
Naturality of the path fibration applied to the map $\eta_X:X\to S|X|$ affords one with a similar diagram contining the second connecting homomorphism. $$ \begin{matrix} \cdots\rightarrow & \pi_{n+1}(X,x) & \xrightarrow{\partial} & \pi_{n}(\Omega_xX,x) & \rightarrow & \pi_{n}(P_xX,x) & \rightarrow & \pi_{n}(X,x) & \xrightarrow{\partial} & \pi_{n-1}(\Omega_xX,x) &\rightarrow\cdots \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ \cdots\rightarrow & \pi_{n+1}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n}(\Omega_xS|X|,x) & \rightarrow & \pi_{n}(P_xS|X|,x) & \rightarrow & \pi_{n}(S|X|,x) & \xrightarrow{{\color{blue}\partial}} & \pi_{n-1}(\Omega_xS|X|,x) &\rightarrow\cdots \end{matrix} $$
Presumably one has to link the two diagrams and do a little diagram chase, but I don't see how.