 Binary-product criterion for lifting-only strong epimorphisms

In a category with binary products in a category, a map with the left lifting property against monomorphisms is an epimorphism. If $ue=ve$, lift the square with right side $\Delta$ and bottom side $⟨u,v⟩$. The two projections of its filler give $u=v$. This argument does not require equalizers.