I read that a morphism $\gamma : S \rightarrow T$ is an isomorphism if there exists a morphism $\Psi : T \rightarrow S$ such that $\gamma \circ \Psi = I(T)$ and $\Psi \circ \gamma = I(S)$, where $I$ denotes the identity.
In the category of semigroups [monoids, groups, semirings], a morphism is an isomorphism if and only if it is bijective.
But after some lines, it was written that the second sentence above does not hold for morphisms of ordered monoids. In particular, if a set $(M,\leq)$ is an ordered monoid with order relationship $\leq$, the identity induces a bijective morphism from $(M,=)$ onto $(M,\leq)$ which is not in general an isomorphism.
I got lost with the above paragraph. What could it mean?