  The monoid of multipliers of a semigroup object in a monoidal category is
introduced, arising from an abstraction of the definition of the translational
hull of an ordinary semigroup or of the multiplier algebra of a Banach algebra
and dually, the monoid of comultipliers of a cosemigroup object is obtained.
Its set-theoretic version, the classical translational hull, is shown to
provide a functor from a subcategory of ordinary semigroups to monoids, similar
to a left adjoint. The abstract multiplier monoid of a semigroup object is
related to the concrete translational hull of its convolution semigroup by a
``concretization'' homomorphism. For semigroup objects for which this
homomorphism is onto, the multiplier construction is functorial and the
concretization homomorphisms form a natural epimorphism.
