bijective homomorphisms between non isomorphic posets, example , explanation needed

Question

Could someone please give a super simple example of a "bijective homomorphism between a non isomorphic poset"?

I don't understand the sentence marked by green in the picture taken from Awodey's book.

Answer 1

Here is a very simple example:

• let $P = \{a,b,c\}$ where $a \le b$ and $a \le c$ ($b$ and $c$ aren't comparable);
• let $Q = \{x,y,z\}$ where $x \le y \le z$.

And let $f : P \to Q$ be defined by $a \mapsto x$, $b \mapsto y$ and $c \mapsto z$. $f$ is obviously bijective, and since $x \le y$ and $x \le z$ it's a poset homomorphism. But it's not an isomorphism, because $Q$ is totally ordered while $P$ isn't.