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This is an exercise given by Vakil’s online notes on algebraic geometry. I wanted to solve the problem by universal properties, but I could not make it complete. I found a solution online as follows:

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This solution cannot convince me when it says (in the second last paragraph), “if the starred dashed arrow is not the identity, then we have two different ways...”. Aren’t both of them zero morphisms? Is this a right solution?

I was also stuck at the last bit where I need to find a reversed arrow of the starred dashed one.

I know this exercise can be solved by other ways such as category theory or the exactness of sheafification, but I wonder how to make this approach work. Thanks!

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