To an extent. The original use of the term "Democratic-Republicans" refers to a number of societies that sprang up after the French revolution in America that held extreme views. These societies generally focused more as a "pressure group" than as political agents but there were exceptions, and ideologically they tended to be closer to Republicans than Federalists, and in fact many society members eventually became loyal Republicans. Jefferson and Madison sympathized with the society but kept their distance fearing their radical ideology and also of being labeled as seditious for supporting them. Important to remember as well that in the early elections there was no party apparatus and candidates ran on literally dozens of different names, at times being quite confusing for instance Federalist-Republicans running against Republican-Federalists. Later some Republicans did adopt the democratic name, largely as a reference to the earlier societies and in an effort to attract new voters to their party. However it should be noted that Federalist and Republican were both much more powerful terms and generally speaking the correct reference is Republican or Jeffersonian-Republican