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I want to find a monic in category OSet defined as

"sets with unary operation, $(A,x)$, where $x:A\rightarrow A$, and morphism preserving that operation, that is a morphism from $(A,x)$ to $(B,y)$ is $f:A\rightarrow B$ with $f\circ x=y\circ f$"

I am doing it exactly how we prove injective functions are monic in category Set, i.e, I prove any injective function $\space$ $m:(A,x)\rightarrow (B,y)$ in OSet is monic by proving for each parallel pair of arrows $f,g:(S,z)\rightarrow (A,x)$, we have$\space$ $m\circ f=m\circ g\implies f=g$.

My confusion is this that do I need to do anything more with it as this time $m$ is not a function but 'operation preserving function'.

Thank you.

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Basically, for a given morphism $m:(A,x)\rightarrow (B,y)$ in $\mathbf{OSet}$ you want to prove

$m$ is monic $\iff$ the underlying map $m:A\rightarrow B$ is injective.

The proof of the $\Leftarrow$ direction works as you described above, just like for $\mathbf{Set}$. But for the proof of the $\Rightarrow$ direction you need to do a little bit more.

What makes the proof work in the case of ordinary sets is the isomorphism $A\cong \mathbf{Set}(1,A)$ (where $1$ is some one-element set), which allows you to view an element $a\in A$ as a map $a:1\rightarrow A$ so that e.g. $m(a) = m(a')$ ''means'' the same as $m\circ a= m\circ a'$.

So you need to identify a suitable object $(N,s)$ in $\mathbf{OSet}$ such that there is an isomorphism $A\cong \mathbf{OSet}((N,s),(A,x)))$. A one-element set (with the identity map) will not work here because maps in $\mathbf{OSet}((1,id),(A,x))$ only correspond to fixpoints of $x$. But perhaps you can guess by my notation a suitable $(N,s)$.

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