  An algebraically exact category in one that admits all of the limits and
colimits which every variety of algebras possesses and every forgetful functor
between varieties preserves, and which verifies the same interactions between
these limits and colimits as hold in any variety. Such categories were studied
by Ad\'amek, Lawvere and Rosick\'y: they characterised them as the categories
with small limits and sifted colimits for which the functor taking sifted
colimits is continuous. They conjectured that a complete and sifted-cocomplete
category should be algebraically exact just when it is Barr-exact, finite
limits commute with filtered colimits, regular epimorphisms are stable by small
products, and filtered colimits distribute over small products. We prove this
conjecture.
