Reading Weibl's Introduction to Homological Algebra we get the definition
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.
I am trying to digest what it means to be universal with respect to a property. From what I understand this means that for any other object satisfying such a property there is a unique isomorphism between the two universal objects. Futher, an object is universal if it is either initial or terminal (and the kernel is clearly an initial morphism with A an initial object?) and the universal property is the universal morphism, that is, the initial or terminal morphism, (which in our case is $i$ which is an initial morphism?). It is my understanding that universal objects and universal properties(/morphisms?) occur together?
I've read the following along with the wikipedia page:
The second is a direct response to the same question but I don't quite find the answer complete enough for my understanding.