Lemma 12.19.10. Let $\mathcal{A}$ be an abelian category. Let $A, B, C \in \text{Fil}(\mathcal{A})$. Let $f : A \to B$ and $g : A \to C$ be morphisms. Then there exists a pushout
in $\text{Fil}(\mathcal{A})$. If $f$ is strict, so is $f'$.
Lemma 12.19.10. Let $\mathcal{A}$ be an abelian category. Let $A, B, C \in \text{Fil}(\mathcal{A})$. Let $f : A \to B$ and $g : A \to C$ be morphisms. Then there exists a pushout
in $\text{Fil}(\mathcal{A})$. If $f$ is strict, so is $f'$.
Proof. Set $C \amalg _ A B$ equal to $\mathop{\mathrm{Coker}}((g, -f) : A \to C \oplus B)$ in $\text{Fil}(\mathcal{A})$. This cokernel exists, by Lemma 12.19.2. It is a pushout, see Example 12.5.6. Note that $F^ p(C \amalg _ A B)$ is the image of $F^ pC \oplus F^ pB$. Hence
Whence the last statement. $\square$
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