By F. Borceux, G. Van den Bossche

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U* = v, w, U* = LL, U* and so for any object A in C we have natural morphisms V, v * ( ~ ) : v , v* A ÷ u , u* A 8U v* : ul u* A-~ v, v* A. v! A But we a l s o have v~ V ~ u! U* = v! w~ U ~ = u! U* • By taking their right adjoints we get u, u* v , v* = u, w* v* = u, u*. So f o r any A in C we have n a t u r a l morphisms 0 v,v v! v * ( ~ ) : v, ~. A + u. u* A A : u! u* A ~ vi v* A. In fact the following equalities hold v, v*(~) u*A V* A = U v, v* A v! v*(~AU) = SU v! v* A 30 Indeed. the following triangular equality arises from the adjunction v* -4 v, * v* V ~ V * Now v!

I s t h e image o f m i n U. and a l e f t T h i s shows t h a t But m i s a monomorphism i n U; so p i s b o t h a rmnomorphism a n d a r e g u l a r e p i m o r p h i s m i n U : i t i s a n isomorphism. So ra i s i s o m o r p h i c t o i , a monomorphisra i n C. Proposition 8. Let C be a category equivalent to some category S h ( H , ~ ) . formal initial segment in C. in C. Let 0 ~ A and i : B ~ Let U be a A be monomorphisms The following square is a pullback in C u(B) > 6B , B , A I u(A) > ~A where the horizontal arrows are those arising from the addunction u!

We first prove that the image of BA, U for any A in C, is the union of the images of the Ui U Ui various BA . Consider the following diagram where BA and BA have been factored through their image : U BA u(A) , A ! 1 "11~ ~ ~uA ui(A) ~A 1 There is a factorization ~i through the images. Thus we obtain the following co~utative diagram for U i ~ U j U. 1 BA U i (A) ~ Pi Ti u U1 ~uj (A) iEI 3 T1 ~j uj(A) U. BA3 So we obtain a cone with vertex q : u(A) ÷ U T i and thus a unique factorization i£I U. l U T.