  G\'alvez-Carrillo, Kock, and Tonks constructed a decomposition space $U$ of
all M\"obius intervals, as a recipient of Lawvere's interval construction for
M\"obius categories, and conjectured that $U$ enjoys a certain universal
property: for every M\"obius decomposition space $X$, the space of culf
functors from $X$ to $U$ is contractible. In this paper, we work at the level
of homotopy 1-types to prove the first case of the conjecture, namely for
locally discrete decomposition spaces. This provides also the first substantial
evidence for the general conjecture.
  This case is general enough to cover all locally finite posets,
Cartier--Foata monoids, M\"obius categories and strict (directed) restriction
species. The proof is 2-categorical. First, we construct a local strict model
of $U$, which is then used to show by hand that the Lawvere interval
construction, considered as a natural transformation, does not admit other
self-modifications than the identity.
