Insertion of lattice-valued functions in a monotone manner is investigated. For L a ⊲-separable completely distributive lattice (i.e. L admits a countable base which is free of supercompact elements), a monotone version of the Katětov-Tong insertion theorem
for L-valued functions is established. We also provide a monotone lattice-valued version of Urysohn’s lemma. Both results yield
new characterizations of monotonically normal spaces. Moreover, extension of lattice-valued functions under additional assumptions
is shown to characterize also monotone normality.