Let $F: \mathcal{A} \to \mathcal{B}$ be an exact covariant functor of abelian categories and let
$$\mathscr{C}: A \to A \to B \to A$$ be an exact couple in $\mathcal{A}$ with corresponding spectral sequence $E^r(\mathscr{C})$.

$$F(\mathscr{C}): F(A) \to F(A) \to F(B) \to F(A)$$
is a exact couple in $\mathcal{B}$ and defines a spectral sequence $E^r(F(\mathscr{C}))$. *I am pretty sure* there is an isomorphism of spectral sequences that is natural in $\mathscr{C}$:
$$E^r(F(\mathscr{C})) \cong F(E^r(\mathscr{C}))$$
Such a isomorphism is equivalent to a natural isomorphism $D(F(\mathscr{C}))\cong F(D(\mathscr{C}))$ for all exact couples where $D(-)$ denotes the derived couple.

**Question:** Does someone know a reference for such an isomorphism or a place in the literature where such an isomorphism is used (i.e. in this generality, not for particular $F)$ ?