We start by recalling the definition of a morphism in the category of Spectral Sequences:

a morphism $f \colon A \to E$ is a family of maps $f^r_{pq}\colon A^r_{pq}\to E^r_{pq}$ in the abelian category $\mathcal{A}$, with $d^rf^r=f^rd^r$ such that each $f^{r+1}_{pq}$ is the map induced by $f^r_{pq}$ on homology.

Now the Mapping Lemma [Weibel page 123, 5.2.4] tells us

Let $A,E$ be spectral sequences in an abelian category $\mathcal{A}$. Let $f \colon A \to E$ be a morphism of spectral sequences such that for some fixed $r$, $$f^r\colon A^r_{pq}\to E^r_{pq}$$ is an isomorphism for all $p$ and all $q$. Then the 5-Lemma implies that $$f^s\colon A^s_{pq}\to E^s_{pq}$$ is an isomorphism for all $s\geq r$ as well

The part I don't understand is the highlighted part.

Why we need 5-lemma? In fact, according to me, if $f^{r+1}_{pq}$ is the map induced in homology, then by hypothesis we have that $f^r_{pq}$ is an isomorphism, therefore it induces in homology an isomorphism and so by induction we are done. Where are (if any) the mistakes in this reasoning?


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