Let J, g : C D be morphisms of a category . For each X in

Let J, g : C → D be morphisms of a category ℓ. For each X in ℓ let (a) Eq( - ,∫,g) is a contra variant functor from ℓ to the category of sets.

(b) A morphism i: K → C is a difference kernel of (f,g) if and only if Eq(- ,→,g) is representable with representing object K (that is, there is a natural isomorphism τ: hom_ℓ(-,K) → Eq(- ,∫,g)). 

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Eq(X, f,g) = {he hom(X,C) | fh = gh}.