Lemma 7 ([Fibers and objects preserve colimits]). For ε∈0,1 , the functor F_ε:/1⟶, (Cp1) C_ε=p^-1 (ε), preserves all small colimits. The object functor Ob: also preserves all small colimits. Both assertions hold directly for n=∞ .

We construct a right adjoint Q_0 to F_0 . Given B , put all its objects over 0 and adjoin a new object * over 1 . Keep the hom categories B(b,b') , and set Q_0(B)(b,*)=1, Q_0(B)(*,b)=∅, Q_0(B)(*,*)=1. Here the homs are strict (n-1) -categories for finite n , sets when n=1 , and strict ω -categories when n=∞ ; 1 and ∅ denote their terminal and initial objects. Compositions inside B are retained. Every remaining composition either has empty domain or has terminal codomain, so is uniquely determined. Associativity, units, and interchange hold by the laws in B or by uniqueness. This defines Q_0(B) and its map to 1 in all the stated dimensions. For p:C→1 , a functor C_0→B extends uniquely over 1 to C→Q_0(B) . All objects of C_1 must go to * ; their internal cells and the cells from C_0 to C_1 have unique images in the specified terminal homs. There are no cells from C_1 to C_0 , since their 1 -boundaries would map to an arrow 1→0 . Restriction and extension are inverse and natural, giving _ /1 (C,Q_0(B)) ≅_ (C_0,B). Thus F_0 is a left adjoint. For F_1 , place B over 1 and adjoin a new initial object over 0 , with terminal homs from that object into B , empty reverse homs, and terminal endomorphism hom. The same restriction argument gives a right adjoint Q_1 . In particular these constructions include B=∅ and prove preservation of the empty colimit as well. For a set S , let K(S) be the ordinary chaotic category on S , with one arrow for each ordered pair of objects, regarded as an n -category with identity cells above dimension one. Every object function ObC→S has a unique strict extension C→K(S) . Consequently Ob is left adjoint to K and preserves colimits.