This will imply that any uniform functor to Set is unique up to equivalence. Consider a left category C with a strict initial 0 . A map is called non-initial if its domain is not initial. We say that f and g are disjoint if the initial object is the pullback of f and g . We say that g dominates f if any non-initial map to X which factors through f is not disjoint with g .
X are three maps.
If g dominates f and h is disjoint with g , then h is disjoint with f . We may assume f is non-initial . This implies that g dominates f . Conversely, assume g dominates f . Since F is uniform, this implies that g is disjoint with s .