I'm studying Category Theory from the book "An Introduction to Category Theory" by Harold Simmons. I found a statement there "Each identity arrow is the corresponding identity function viewed as a monotone map." Can anyone kindly explain it to me what it means? Thanks!
P. S. I am NOT asking about identity function viewed as a morphism, which I have already proven.
In the category of posets, the objects are posets (which are just sets with some ordering) and the morphisms are monotone functions. What Simmons wants to say is that a ordinary, set-theoretic, identity map $R\to R$ is a monotone map when $R$ has an ordering, because the reflexivity of a poset. So he says: 'The identity arrow on an object $R$ in the category of posets is the same as the identity arrow on a set $R$ in the category of sets, but now interpreted as a monotone map.' Hope this helps.