**Proof.**
It is clear that $\text{Fil}(\mathcal{A})$ is additive with direct sum given by $(A, F) \oplus (B, F) = (A \oplus B, F)$ where $F^ p(A \oplus B) = F^ pA \oplus F^ pB$. The kernel of a morphism $f : (A, F) \to (B, F)$ of filtered objects is the injection $\mathop{\mathrm{Ker}}(f) \subset A$ where $\mathop{\mathrm{Ker}}(f)$ is endowed with the induced filtration. The cokernel of a morphism $f : A \to B$ of filtered objects is the surjection $B \to \mathop{\mathrm{Coker}}(f)$ where $\mathop{\mathrm{Coker}}(f)$ is endowed with the quotient filtration. Since all kernels and cokernels exist, so do all coimages and images. See Example 12.3.13 for the last statement.
$\square$

## Comments (0)

There are also: