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.