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In Paolo Aluffi's Algebra: Chapter 0, there is an exercise which asks

Prove that there is a surjective homomorphism (an epimorphism) from $\Bbb Z*\Bbb Z$ onto $C_2*C_3$, where $A*B$ is the coproduct of $A$ and $B$ in $\mathsf{Grp}$. One can think of $\Bbb Z*\Bbb Z$ as a group with two generators $x, y$ subject to no relations whatsoever.

I know that this question has been asked previously here and here. But neither of them have correct answer that uses the machinery that has already been introduced. If you have not read the book, try not to use anything about coproducts in the answer other than the fact that they are initial objects in the category $C^{A,B}$.

Here's what I thought:

I know that since $\Bbb Z*\Bbb Z$ is an initial object, there is a unique morphism from $\Bbb Z*\Bbb Z$ onto $C_2*C_3$ in the category $C^{A,B}$. The only question is how to prove surjectivity of the morphism. We can construct morphisms from $\Bbb Z \to C_2 \text{ and } C_3$ that are surjective. Somehow we need to compose the two to form the required morphism.

Rohit Gupta
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    "...there is a unique morphism $\Bbb Z\Bbb Z$ onto $C_2C_3$." Something important is missing from this description. – Chris Culter Sep 30 '17 at 08:41
  • Other than the word 'from'? – Rohit Gupta Sep 30 '17 at 08:46
  • Instead of marking the question as duplicate, you could have noticed that the answer given there is not correct. It uses information about $C_{2}*C_{3}$ that is not available at the time of question. – Rohit Gupta Sep 30 '17 at 12:25

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