The Stacks project

Remark 12.3.6. Note that the proof of Lemma 12.3.4 shows that given $p$ and $q$ the morphisms $i$, $j$ are uniquely determined by the rules $p \circ i = \text{id}_ x$, $q \circ j = \text{id}_ y$, $p \circ j = 0$, $q \circ i = 0$. Moreover, we automatically have $i \circ p + j \circ q = \text{id}_{x \oplus y}$. Similarly, given $i$, $j$ the morphisms $p$ and $q$ are uniquely determined. Finally, given objects $x, y, z$ and morphisms $i : x \to z$, $j : y \to z$, $p : z \to x$ and $q : z \to y$ such that $p \circ i = \text{id}_ x$, $q \circ j = \text{id}_ y$, $p \circ j = 0$, $q \circ i = 0$ and $i \circ p + j \circ q = \text{id}_ z$, then $z$ is the direct sum of $x$ and $y$ with the four morphisms equal to $i, j, p, q$.

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