  It is shown that every continuous homomorphism of Arens-Michael algebras can
be obtained as the limit of a morphism of certain projective systems consisting
of Fr\'{e}chet algebras. Based on this we prove that a complemented subalgebra
of an uncountable product of Fr\'{e}chet algebras is topologically isomorphic
to the product of Fr\'{e}chet algebras. These results are used to characterize
injective objects of the category of locally convex topological vector spaces.
Dually, it is shown that a complemented subspace of an uncountable direct sum
of Banach spaces is topologically isomorphic to the direct sum of ({\bf
LB})-spaces. This result is used to characterize projective objects of the
above category.
