The Stacks project

Lemma 12.3.18. Let $\mathcal{A}$ be a preadditive category, $x$ and $y$ be objects of $\mathcal{A}$, and $j : y \to x$ and $q : x \to y$ be morphisms satisfying $q \circ j = \text{id} _ y$. If $j \circ q$ has at least one of a kernel or cokernel, then both (co)kernels exist and

\begin{align*} x & \simeq \mathop{\mathrm{Ker}}(j \circ q) \oplus y \\ & \simeq \mathop{\mathrm{Coker}}(j \circ q) \oplus y \end{align*}

with the structure of either direct product including $j , q$, and the canonical (co)kernel morphism, and the remaining map uniquely determined.

Proof. Write $f = j \circ q$, compute that $f \circ f = f$, and apply Lemma 12.3.16 and Lemma 12.3.17. $\square$

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