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T h e small category we obtained in this way will be considered as the diagram scheme for our diagram. If the diagram scheme consists of two objects X and Y and of three morphisms \ > 1 , and x : X —• Y, then we call this category 2. T h e diagrams of Funct(2, ? ) are in one-one correspondence to the morphisms of €. T h u s one calls Funct(2, ? ) the morphism category of ? A morieJ c x c F 26 1. PRELIMINARY NOTIONS phism in Funct(2, ? 5, we want to generalize again a notion from S to arbitrary categories.

A l l surjective set maps are retractions. I n A b the map Zsn\-> 2n e Z is a kernel of the residue class homomorphism Z —>- Z/2Z; however, it is not a section. In fact, if g : Z —>- Z were a corresponding retraction, then 2^(1) — 1 e Z. But there is no such element ^(1) in Z. LI 1 Products and Coproducts Another important notion in the category of sets is the notion of a produet of two sets A and B. T h e produet is the set of pairs A x B = {(a b)\aeA t and b e B} 30 1. PRELIMINARY NOTIONS Furthermore, there are maps p : A x 5 9 (a, 6) H-> ae A A and p : A X B3{a b)\-> B y be B We want to investigate whether this notion can again be generalized i n the desired way to morphism sets.

If each [finite, nonempty] family of objects i n ? has a produet, then we call ? a category, with \ßnite nonempty] produets. If (A {pu}) is a produet of a y } 32 1. PRELIMINARY NOTIONS family of objects {A^ i n ^ and if h : B —> A is an isomorphism, then (B, { pih}) is another produet for the A . i£l i L E M M A 2. Assume that in the category there is a produet for each pair of objects. Then is a category with finite, nonempty produets. Proof. L e t A A be a family of objects i n ? W e show that (••• (A X A ) X •*•) X A is a produet of the A .