Define a monomorphism (epimorphism) as a left-cancellative (right-cancellative) morphism, in some category.
The Wikipedia page on cokernels says that a morphism is injective (surjective) iff its kernel (cokernel) is trivial. The category being discussed seems to be that of vector spaces.
Now, in many categories, being an injection (surjection) is equivalent to being a monomorphism (epimorphism), but, if this is not the case --ie. monomorphisms are not injective (epimorphisms are not surjective)--, does the above result generalize to such monomorphisms (epimorphisms)?
Namely, is the following true? A morphism is a monomorphism (epimorphism) iff its kernel (cokernel) is trivial.