Lemma 12.24.4. Let $\mathcal{A}$ be an abelian category. Let $\alpha : (K^\bullet , F) \to (L^\bullet , F)$ be a morphism of filtered complexes of $\mathcal{A}$. Let $(E_ r(K), d_ r)_{r \geq 0}$, resp. $(E_ r(L), d_ r)_{r \geq 0}$ be the spectral sequence associated to $(K^\bullet , F)$, resp. $(L^\bullet , F)$. The morphism $\alpha $ induces a canonical morphism of spectral sequences $\{ \alpha _ r : E_ r(K) \to E_ r(L)\} _{r \geq 0}$ compatible with the bigradings.

**Proof.**
Obvious from the explicit representation of the terms of the spectral sequences.
$\square$

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