By F. Borceux, G. Van den Bossche

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**Example text**

We need to produce a morphism YF ~'F : O F F = a O' F F ÷ F which is equivalent, by adjunction, to a morphism YF : PF÷F So, if U is some formal initial seg~nt, we must find a homomorphism ¥~(U) : lira F ( v ) ÷ F U; v~U v£I~ Take i t t o be t h e f a c t o r i z a t i o n F(v) + F U. e. map a morphism o f p r e s h e a - ves, Now consider A in Sh(14,1~) and a morphism f : OA ÷ F in Sh(H,~). We must find a unique factorization g : A ÷ F F such that YF ° o(g) = f. 0 P F = a O' r F ..... ogla O' g / YF ~ F f oA=ao'A If such a g exists, evaluate this diagram at v C ~ and get 48 Fv Fv A(v) " This shows uniqueness of g.

F 8) if 0 ÷ M is a monomorphism in U, u! preserves monomorphisms with codo- main M. As u I is full and faithful so is u. (cfr. [2]] - ]6 - 8 - 9). Also u I reflects monomorphisms because it is faithful. Now consider a monomorphism m : N ÷ M in U with 0 ÷ M a monomorphism in U. Consider the image of m in C N By (F 3 ) , I is is invariant adjoint; I still m ~ M (up t o an isomorphism) i n U; t h u s t h e whole d i a g r a m i s i n U and u n d e r u* ( c f r . [21] - ]6 - 5 - 4 ) . But u* h a s a r i g h t so i t p r e s e r v e s monomorphisms and r e g u l a r e p i m o r p h i s m s .

U*(a F hl) = a F h u. if v~__
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