How to understand "unique" for universal properties

Question

I'm reading this basic text on Category Theory. The author presents the concept of a universal property early, and I'm wondering about one of the examples. The author writes

Example 0.2 This example involves rings, which in this book are always taken to have a multiplicative identity, called $1$. Similarly, homomorphisms of rings are understood to preserve multiplicative identities. The ring $\mathbb{Z}$ has the following property: for all rings $R$, there exists a unique homomorphism $\mathbb{Z} \to R$

The author then proceeds with a construction of the claimed homomorphism, and states that this is a universal property for rings, in the sense that if this property is satisfied for any ring $R$ then $\mathbb{Z} \cong R $

I'm not sure how to feel about this, because, taken at face value, it seems that is states that $\mathbb{Z}$ has the property that, given any other ring $R$, there exists one and only one homo $\mathbb{Z} \to R$, but this is not true! Taking $R = \mathbb{Z}$, this is not true by this question.

Maybe the author means something else? Or how does this work?

Thanks in advance

Answer 1

There is a unique homomorphism $\mathbb{Z}\to R$ for any ring $R$. The question you link to is about group homomorphisms.

Ring homomorphisms are additive group homomorphisms which furthermore respect multiplication and map the unit element of one ring to the other. In the case of $\mathbb{Z}$, this clearly forces the image of all other elements as well.

Answer 2

There is a unique such homomorphism of 'unital rings', i.e. $1$ as a constant is considered to be part of the structure, and hence has to be mapped to $1$ under a ring homomorphism.