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M o r ^ Z ) , 9Y(D)) 3 if left adjoint to 9 : 3 -> # if and only if there are natural transformations 0 : \d<# -> 9 ^ and W : -> \d with = i d and (W&){&&) = i d ^ . @ y COROLLARY 4. Let 3F be left adjoint to 9, then the maps 9 : Mor^(jFC, D) are injective for all C and D e 3.

F~ {B ) = h(B ) if f is an isomorphism with the inverse morphism h. f~\g-\C )) = (gf)~\C ) ifboth sides are defined. g(f(A )) = (gf)(A ) ifboth sides are defined, iff(A ) andg(f(A )) are epimorphic images, and if ? is balanced. (h) f(A ) =ff~ f(A ) iff-JiAJ is defined. (i) f-^BJ = / - y / - i ( f i ) ifff-\B£ « defined. ) = f(\J A ) if \J A is defined and ? is a category with images and coimages. ) if the right side is defined. x 2 1 2 1 x 1 x l x x x x } x x x 1 t 1 1 i€I t i iGl Proof. T h e assertions (a)-(e) arise directly from the corresponding definitions.

Let / : A —* B be a morphism i n ? and ^ : fi' B be a subobject of B. A subobject yJ' -> A of is called a counterimage of 5 ' under / if there is a morphism f':A'—> B' such that the diagram j i« is commutative and if for each commutative diagram C 1 >B' V there is exactly one morphism h : C —*• Ä such that the diagram C 36 1. PRELIMINARY NOTIONS is commutative. T h i s condition asks for more than that A' be only the largest subobject of A which may be transferred by / into B'. But the condition implies this assertion.

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Categories and functors by Bodo Pareigis

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