Given an object in some category, a family of morphisms to is an epic sink, or a jointly epic family if, given any two morphisms such that for all , it follows that .
Dually, a family of morphisms from is a monic source, or a jointly monic family if, given any two morphisms such that for all , it follows that .
Sometimes we are interested only in small families of morphisms, but if so then it is best to say so explicitly.
A single morphism is an epimorphism if and only it forms an epic sink by itself; conversely, a sink is epic iff the induced map is an epimorphism, assuming that the coproduct exists. (Note, though, that for a large family of morphisms, this coproduct might not exist even if the category has all small coproducts.) Dual results hold for monomorphisms and products.