In the definition of categories, one of its condition :for any three objects A,B & C and its morphisms we have the composition of the two morphisms which takes from A to C Does that mean that we don't always have a composition between morphisms or it means that the composition must be in the set Mor(A,C) that is associated with the category.
Objects and its morphisms
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abstract-algebra
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1 Answers
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What must hold in any category, is that if you have a morphism $f:A \to B$ and a morphism $g: B \to C$, then there must be a morphism $g \circ f: A \to C$.
But this doesn't mean that you must have morphisms between any two objects: Take any set $S$ and form the category whose objects are the elements of the set, and whose only morphisms are the identity morphisms $id:s \to s$ for any $s \in S$.
So to summarize: if you have two morphisms, one with the same target as the domain of the other, then there is always a composition between them.