Local theory extensions provide a complete and efficient way for reasoning about satisfiability of certain kinds of universally quantified formulas modulo a background theory. They are therefore useful in automated reasoning, software verification, and synthesis. In this paper, we 1) provide a sufficient condition for locality of axioms with one variable specifying functions and relations, 2) show how to obtain local axiomatizations from non-local ones in some cases, 3) show how to obtain piecewise combination of different local theory extensions.