Spectral sequence of a filtered cochain complex

Definition
Let $$\mathcal{A}$$ be an abelian category. We describe below what a filtered cochain complex over $$\mathcal{A}$$ means and what it means to be a spectral sequence over such a filtered cochain complex.

Setup

 * $$C^\cdot$$ is a cochain complex over $$\mathcal{A}$$. In other words, it has objects $$C^q, q \in \mathbb{Z}$$, with morphisms $$C^q \to C^{q+1}$$ such that the composition of two successive morphisms of this sort gives the zero morphism.
 * We are given a filtration of subcomplexes $$F^pC^\cdot$$ where $$p \in \mathbb{Z}$$. Each $$F^pC^\cdot$$ is a subcomplex of $$C^\cdot$$, and the filtration is descending, i.e., $$F^pC^\cdot \supseteq F^{p+1}C^\cdot$$.