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Math Help - Subobjects in a category

  1. #1
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    Subobjects in a category

    In a category \mathcal{C} a subobject of an object A is an object B such that there exists a monic \phi : A \longrightarrow B. Simple enough.

    But I need to show that if A is a subobject of B and B is a subobject of A, then A = B.

    It looks obvious but I think I am missing something.
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  2. #2
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    Ah, If \theta : B \longrightarrow A is the other monic then their composition is monic, but their compositions are identity morphisms for A and B; \theta \circ \phi = 1_A and \phi \circ \theta = 1_B. Hence they are isomorphic! Or is there a strict equality? help
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