Problem:
If $X^{\circlearrowleft \alpha}$ is any object of $S^\circlearrowleft$ for which there exists an injective $S^\circlearrowleft$-map $f$ to some $Y^{\circlearrowleft \beta}$ where $\beta$ is in the subcategory of automorphisms, then $\alpha$ itself must be injective.
The definition injectivity the book:
- A map $X \xrightarrow{a} Y$ is injective iff for any maps $T \xrightarrow{x_1} X$ and $T \xrightarrow{x2} X$ (in the same category) if $a \circ x_1=a \circ x_2$ then $x_1=x_2$.
The definition of $S^{\circlearrowleft}$ is: A category whose objects are sets equipped with endomaps.
I believe this question claims that we need to show injectivity in the category of endomaps where they say:
If x1 and x2 were not only S-maps, but also S↺-maps (for a given endomap τ on T), by the injectivity of f, we'd get x1=x2. However, this may not be the case.
I don't know if that is right. My interpretation of the question is that we want to show $\alpha$ is injective in the category of sets because by definition $\alpha$ is simply a set map $\alpha\colon X \to X$. Is one interpretation more appropriate than the other?
Additionally, the hints in the other question seem like they want to invoke the universal property of the natural numbers (is this correct?), which is something the textbook won't even cover for another hundred pages or so.
To easily prove that an injective $S^\circlearrowleft$ map is also an injective $S$-map requires the Yoneda lemma or other tools more advanced than I believe the reader is expected to have at the given point in the book. So it's reasonable, especially since $f$ being injective in either category really is equivalent, to assume $f$ is injective as an $S$-map and proceed from there.