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

\[ \xymatrix{ A \ar[r]_ f \ar[d]_ g & B \ar[d]^{g'} \\ C \ar[r]^{f'} & C \amalg _ A B } \]

in $\text{Fil}(\mathcal{A})$. If $f$ is strict, so is $f'$.

**Proof.**
Set $C \amalg _ A B$ equal to $\mathop{\mathrm{Coker}}((1, -1) : A \to C \oplus B)$ 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

\[ (f')^{-1}(F^ p(C \amalg _ A B)) = g(f^{-1}(F^ pB))) + F^ pC \]

Whence the last statement. $\square$

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