Free comonoids in models of linear logic

The construction of the free monoid by means of words is a very old
practice in mathematics. Once recognized the categorical nature of the
concept of monoid, a lot of similar free constructions in mathematics
(like tensor algebras, differential graded tensor algebras, free
categories, etc) become unified by the same formula inspired by the
construction in Set.  This formula works because the underlying
categories have coproducts and a tensor product which commutes with
them.
We have learned from linear logic that the dual notion of comonoid
captures in essence the discharge and copy mechanisms in proofs and
programs.  It is therefore natural to wonder whether a comonoid !A may
be obtained as a free construction from a given object
A. Unfortunately, most of the categories giving rise to models of
linear logic do not satisfy the dual of the property stated above:
their tensor product does not commute with their cartesian product.
In this talk, we explain how to deduce a general construction of the
free monoid in a monoidal category from the construction of a free
pseudomonoid in the surrounding 2-category of categories. This leads
us to an alternative formula, in which the free commutative comonoids
is computed as a particular categorical limit.  We show that the
formula works in two extremely different models of linear logic: the
category of coherence spaces, in which it reconstructs the well-known
modality !A based on multi-cliques; the category of Conway games, in
which it generates the free commutative comonoid !A by indexing the
copies of the game A in an incremental way.