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In this blog post, we will discuss a category theory question, in the framework of homomorphisms of abelian groups.

Let be a homomorphism of abelian groups. Suppose that is a homomorphism of abelian groups such that is the zero map. (One example is the inclusion )

Are the following true or false?

(i) There is a unique homomorphism such that .

(ii) There is a unique homomorphism such that .

It turns out that (i) is false. We may construct a trivial counterexample as follows. Consider , and . Let , be both the zero maps. Then certainly . . Then, for any , , and hence is not equals to the the inclusion map .

It turns out that (ii) is true, in fact it is the famous universal property of the kernel, that any homomorphism yielding zero when composed with has to factor through .

First we will prove **uniqueness.** Let , where is another such map with the property (ii). Then for all , , which implies . Since is the inclusion map, this means that and thus .

Next, we will prove **existence**. Consider . Note that by definition thus .

Next we prove it is a homomorphism. .

Finally by construction it is easy to see that for all .