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Two important consequences of the functor axioms are:

* F transforms each commutative diagram in C into a commutative diagram in D;
* if f is an isomorphism in C, then F(f) is an isomorphism in D.

One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor ''G'' ? ''F'' from A to C. Composition of functors is associative where defined. Identity

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