The reduced Lefschetz number, that is, L ( ⋅ ) − 1 where L ( ⋅ ) denotes the Lefschetz number, is proved to be the unique integer-valued function λ on self-maps of compact polyhedra which is constant on homotopy classes such that (1) λ ( f g ) = λ ( g f ) for f : X → Y and g : Y → X ; (2) if ( f 1 , f 2 , f 3 ) is a map of a cofiber sequence into itself, then λ ( f 1 ) = λ ( f 1 ) + λ ( f 3 ) ; (3) λ ( f ) = − ( deg ( p 1 f e 1 ) + ⋯ + deg ( p k f e k ) ) , where f is a self-map of a wedge of k circles, e r is the inclusion of a circle into the r th summand, and p r is the projection onto the r th summand. If f : X → X is a self-map of a polyhedron and I ( f ) is the fixed point index of f on all of X , then we show that I ( ⋅ ) − 1 satisfies the above axioms. This gives a new proof of the normalization theorem: if f : X → X is a self-map of a polyhedron, then I ( f ) equals the Lefschetz number L ( f ) of f . This result is equivalent to the Lefschetz-Hopf theorem: if f : X → X is a self-map of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f .