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!(h(t)) =!(? f (a) ? S 1 (g(t)) ? ? f (b) ) ,
Now let l : K ? SL be another functor such that { } ? l = g and ! ? l = f . we have l(t) ? SL[l(a), l(b)] the structure theorem gives us: l(t) = ? l(a) ? S 1 ({l(t)}) ? ? l(b) = ? f (a) ? S 1 (g(t)) ? ? f (b) . So l = h. The diagram is a pullback square. Proof of Lemma 7. By construction C p is the pullback of, f (b) = f (t) ,
, We consider the wire striper functor { } : SL ? L. It is clearly essentially surjective. It is also full and faithful since it induces a bijection between SL
, ? b )) ? ? b = ? a ? B(O(? )) ? ? b . Thus we only need to show that B(O(? )) of BoxP of type n · 1 ? m · 1 is in boxed form if it is of the form ? m ? f ? ? n . A diagram in boxed form satisfies B(O(? )) = ? thus we just have to show that any ? : n · 1 ? m · 1 can be rewrite into boxed form
, The structure theorem of BoxL finally gives us ? = ?. U nbox is faithful. Given ? ? SL[a, b], the structure theorem gives us ? = ? b ? S 1 ({?}) ? ? a Then U nbox(? b ? B({?}) ? ? a ) = ? b ? S 1 (O(B({?}))) ? ? a =, Proof of Lemma 9. U nbox is an N 0 -colored prop morphism so it is essentially surjective