Limits and colimits exist in LSet . Thus fibre products exist in LSet . Hence limits exist in LSet . Clearly the union of a set of local ringed sets is the coproduct of the local ringed sets. Thus to prove that colimits exist in LSet it suffices to prove that cokernels exist in LSet .
Y are two morphisms of local ringed sets.
We obtain a ringed set X . O z is local. Thus we obtain a local ringed set X which is clearly the cokernel of f and g in LSet . O is closed under rational operations. Any section of X which is the union of the elements of a subcollection of O is in O .