To have a category, the elements and morphisms have to satisfy some constraints:
<pre>
- (i) the class of morphisms has to be closed under composition:
- where f:C1→C2 and g:C2→C3, g ∘ f is also a
- morphism of the category, which maps C1→C3.
- (ii) composition of morphisms has to be associative
+ (i) the class of morphisms has to be closed under composition:
+ where f:C1→C2 and g:C2→C3, g ∘ f is also a
+ morphism of the category, which maps C1→C3.
+
+ (ii) composition of morphisms has to be associative
+
(iii) every element E of the category has to have an identity
- morphism 1<sub>E</sub>, which is such that for every morphism
- f:C1→C2: 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
+ morphism 1<sub>E</sub>, which is such that for every morphism
+ f:C1→C2: 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
</pre>
These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism.