  The inner automorphisms of a group G can be characterized within the category
of groups without reference to group elements: they are precisely those
automorphisms of G that can be extended, in a functorial manner, to all groups
H given with homomorphisms G --> H. Unlike the group of inner automorphisms of
G itself, the group of such extended systems of automorphisms is always
isomorphic to G. A similar characterization holds for inner automorphisms of an
associative algebra R over a field K; here the group of functorial systems of
automorphisms is isomorphic to the group of units of R modulo units of K.
  If one substitutes "endomorphism" for "automorphism" in these considerations,
then in the group case, the only additional example is the trivial
endomorphism; but in the K-algebra case, a construction unfamiliar to ring
theorists, but known to functional analysts, also arises.
  Systems of endomorphisms with the same functoriality property are examined in
some other categories; other uses of the phrase "inner endomorphism" in the
literature, some overlapping the one introduced here, are noted; the concept of
an inner {\em derivation} of an associative or Lie algebra is looked at from
the same point of view, and the dual concept of a "co-inner" endomorphism is
briefly examined. Several questions are posed.
