We find topological conditions on a space X under which the left (right, or two-sided) uniformity of the free topological group F(X) induces the universal uniformity
or the product uniformity ux × ux on the square of X. Special attention is given to kω metrisable spaces. The main technical tool in the paper is an extension of certain continuous pseudometrics from X2 to F(X) considered by the author in the previous volume of this journal.