Right-factor cancellation for lifting-only strong morphisms

If $gf$ has the left lifting property against monomorphisms, so does $g$. Precompose a lifting square for $g$ with $f$ and lift the composite. To check the remaining triangle, cancel the monic side of the square, rather than cancelling $f$. In particular, a monic right factor of a lifting-only strong morphism is an isomorphism.