Given two local compositions 
, a morphism 
is a couple 
, consisting of a morphism of presheaves 
and an address change 
such that there is a form morphism 
which makes the diagram of presheaves
commute. This defines the
category 
of local compositions. If both,
are objective with
, one may also define morphisms on the sets
by the expressions
(forgetting about the names) which means that
is a set map such that there is a form morphism
which makes the diagram
of sets commute. This defines the
category 
of objective local compositions. Every objective morphism
induces a functorial morphism
in an evident way. This defines a functor
This functor is fully faithful. Moreover, each functorial local composition
(again forgetting about names) gives rise to its
objective trace 
where
. If we fix the address
and restrict to the identity
as address change, we obtain subcategories
and a corresonding fully faithful embedding
. In this context, the objective trace canonically extends to a left inverse functor
of
. Moreover
For more algebraic calculations, such as Grothendieck topologies and 
ech cohomology, one has to restrict to special subcategories. We shall therefore also look at the »address« category 
of left 
-modules with 
-affine morphisms for a given commutative ring 
. In this category, the set of morphisms from module 
to module 
is denoted by 
. We denote the corresponding category of (objective) local compositions by 
(
) Proposition 2 is also valid mutatis mutandis for 
.
3.2 Finite Completeness
So, on a fixed address, objective and associated functorial local compositions are quite the same. But there is a characteristic difference when allowing address change. This relates to universal constructions:
