I think this is a silly question, but I'm quite confused. In Lurie's "Rotation Invariance In Algebraic K-Theory" Notation 3.2.4. he defines a filtered spectrum $\mathbb{A}$ given by $$\mathbb{A}_n = \begin{cases} \mathbb{S} &\text{ if } n =0 \\ 0 &\text{ otherwise}\end{cases} $$
He then notes (example 3.2.9) that $$\mathbb A \otimes \mathbb A$$ takes the form $$ \cdots \to 0 \to 0 \to \mathbb S \to \Sigma \mathbb S \to 0 \to \cdots $$ It seems to me like this should (as a filtered spectrum) be the same as $\mathbb A + \Sigma \mathbb A(1)$, since there are no nontrivial maps $\mathbb S \to \Sigma \mathbb S$. But later (Construction 3.4.4) he writes a fiber sequence $$ \mathbb A \otimes \mathbb A \to \mathbb A \to_\beta \Sigma^2 \mathbb A (1) $$ and gives $\beta$ a name ("the anchor map"). But $\beta$ corresponds to a square $\require{AMScd}$ \begin{CD} \mathbb{S} @>>> 0\\ @VVV @VVV \\ 0 @>>> \Sigma^2\mathbb S \end{CD} and therefore a map of spectra $\mathbb S \to \Sigma \mathbb S$, and therefore (to my eyes) should equal zero.
What am I missing here? If this map is zero, why does it get a name? And if it's nonzero, why doesn't this produce a nontrivial element of $\pi_{-1}\mathbb{S}$?
