  Two first-order logic theories are definitionally equivalent if and only if
there is a bijection between their model classes that preserves isomorphisms
and ultraproducts (Theorem 2). This is a variant of a prior theorem of van
Benthem and Pearce. In Example 2, uncountably many pairs of definitionally
inequivalent theories are given such that their model categories are concretely
isomorphic via bijections that preserve ultraproducts in the model categories
up to isomorphism. Based on these results, we settle several conjectures of
Barrett, Glymour and Halvorson.
