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In an additive category $\mathcal{A}$ a kernel of a morphism $f: B\to C$ is defined to be a map $i : A \to B$ such that $fi = 0$ and that is universal with respect to this property.

This is quoted from Weibel's Homological Algebra. What do they mean by universal with respect to and is $i$ a morphism or a set map? I think it means that if there exists another $j : A'\to B$ such that $fj = 0$, then there is a unique homomorphism $\varphi$ from $A \to A'$ such that $j \varphi = i$, but I'm not sure.

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    $\begingroup$ @MikeMiller That's an answer. $\endgroup$ – Kevin Carlson Jun 14 '15 at 22:36
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All maps are morphisms (unless otherwise stated) - you're in an arbitrary additive category, where set maps don't even make sense.

By universal, it means that any other $j$ that satisfies that property (i.e., any $j: A' \to B$ with $fj = 0$) factors uniquely through $i$ (i.e., there is a unique map $g: A' \to A$ with $j = ig$.) This is what one means by universal given any condition: given anything else that satisfies the condition, it factors uniquely through your universal thing.

It is helpful to draw the diagrams, and to prove that this is true for a category whose kernels one knows well (say, Ab).

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