# Category Theory - Morphisms, Uniqueness & Equivalence

If f:A-->B is an equivalence in a category C and g:B-->A is a morphism such that gf=1_A (the identity on A) and fg=1_B, show that g is unique.

Prove that any two universal (initial) objects in a category C are equivalent.

Prove that any two couniversal (terminal) objects in a category C are equivalent.

In the category of abelian groups, show that the group A_1 x A_2 together with the homomorphisms f:A_1 --> A_1 x A_2, f(x)=(x,e) and g:A_2 --> A_1 x A_2, g(x)=(e,x) is a coproduct for {A_1,A_2}. Why is this not a coproduct in the category of groups?

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#### Solution Summary

Category equivalence is investigated.

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