  A C*-metric algebra consists of a unital C*-algebra and a Leibniz Lip-norm on
the C*-algebra. We show that if the Lip-norms concerned are lower
semicontinuous, then any unital *-homomorphism from a C*-metric algebra to
another one is necessarily Lipschitz. It results that the free product of two
Lipschitz unital *-homomorphisms between C*-metric algebras coming from
*-filtrations is still a Lipschitz unital *-homomorphism.
