A locally finite variety is said to be homogeneous if every isomorphism between subalgebras of a finite algebra in the variety extends to an automorphism of the algebra. The author proves the following claim: every homogeneous locally finite variety of finite type is finitely axiomatizable. As a consequence, the main result of a forthcoming paper [\textit{M. Valeriote} and the author, ``Discriminating varieties'', Algebra Univers. (to appear)] provides a complete structural characterization of the homogeneous locally finite varieties of finite type.