We say U has free objects if the free objects on any set exists. Any bijective morphism in A is an isomorphism. Any pair of parallel morphisms in A has a surjective coequalizer. An algebraic functor U is finitary if it preserves direct colimits. Definition 2 .
Any object has a unit difference.
The product of any two objects is co-universal . Any difference of an object is invertible. The image of an idempotent under any morphism is an idempotent. Any difference of an object has a localization. The sixth axiom simply means that the opposite of the finitary algebraic geometry is a strict coherent analytic geometry .