  The homomorphic image of a congruence is always a tolerance (relation) but,
within a given variety, a tolerance is not necessarily obtained this way. By a
Maltsev-like condition, we characterize varieties whose tolerances are
homomorphic images of their congruences (TImC). As corollaries, we prove that
the variety of semilattices, all varieties of lattices, and all varieties of
unary algebras have TImC. We show that a congruence n-permutable variety has
TImC if and only if it is congruence permutable, and construct an idempotent
variety with a majority term that fails TImC.
