How abusive of notation is it to drop isomorphisms?

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I have a general sort of structural question. I have been reading a lot of maths papers lately, and it seems there are some isomorphisms that people omit from their calculations. For example, in a category with a terminal object, 1,
A \cong A \times 1
where the isomorphism is given from left to right by \langle 1_A , !_A \rangle where ! is the unique map into the terminal object, and the isomorphism from right to left is (left) projection. Now, let a : 1 \to A; an example of the abuse of notation I have seen quite often is to regard
\langle a , 1_1 \rangle : 1 \to A \times 1 as just a

Are there any obvious problems with making such an association? Are there any non-obvious problems with making such an association? Am I missing something?
 
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Mathematics would be much more difficult to read if we didn't make such identifications. Usually, there are no problems with it, but you should always keep in mind that you made the identification. Sometimes you can find yourself very confused about something, and it turns out you forgot you identified some things previously. But normally, there are no problems...
 
Thank you for the reply!
 

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