If is a universal algebra, the set of endomorphisms of forms a monoid (i.e., semigroup with identity) under composition. We denote it by End (). For definitions and notations, see [1]. It is well known (e.g., [1], Theorem 12.3) that for any monoid M there is a unary algebra with M ≅ End (). E. Mendelsohn and Z. Hedrlin [3] have proved that the monoid of a subalgebra of an algebra is independent of the monoid of . In [2], Hedrlin proves the same for the monoid of a homomorphic image of . The proofs of these depend heavily on graph-theoretical and category-theoretical considerations. In this note considerably shorter direct algebraic proofs are given of these results.