Birkhoff variety theorem

I have a vague idea that Michael Barr has a result which answers the following question: that being to give characterization to that situation when a composite of functors A0 -----> A -----> X is tripleable, assuming A -----> X is, and that A0 -----> A is the inclusion of a subcategory. Perhaps the result reads like ``if and only if there exists a Horn clause....''. Might anyone give me a reference?