What is the coproduct: A + A?
The coproduct is the disjoint union. Set-theoretically, you can think of forming the coproduct of the sets $A$ and $B$ as: 

$$
 A + B \;\;\triangleq\;\; \{ (0, a) \;|\; a \in A \} \cup \{ (1, b) \;|\; b \in B \}
$$

Now it should be obvious that $A + A \not= A$.