By Yoneda lemma, kernel of morphism in preadditive category is unique

Question

I'm reading https://stacks.math.columbia.edu/tag/09SE.

It says

How do we use Yoneda lemma to prove the uniqueness of the kernel of a morphism? I tried this:

Please help, thanks.

Answer 1

An important corollary of Yoneda Lemma is that the Yoneda embedding is full and faithfull. Hence any two objects representing the same functor must be isomorphic.