I am having trouble with the following lemma from J. Rotman's Galois Theory book:
Lemma $83$. If $L$ and $L'$ are lattices and $\gamma: L \rightarrow L'$ is an order reversing bijection $[a \leq b$ implies $\gamma(b) \leq \gamma(a)]$, then $\gamma(a \wedge b) = \gamma(a) \vee \gamma(b)$ and $\gamma(a \wedge b) = \gamma(a) \vee \gamma(b)$.
The proof is fairly obvious if you know that when $\gamma$ is order reversing, then $\gamma^{-1}$ is order reversing also. Rotman says that this is "easily seen", but I just don't see it. I think it would be easy if his definition was that $\gamma$ is order reversing when $a \leq b$ if and only if $\gamma(b) \leq \gamma(a)$ instead. In a set theory book I have, isomorphisms between two posets $A$ and $B$ are defined as $f: A \rightarrow B$ such that $a \leq b$ if and only if $f(a) \leq f(b)$.
Am I missing something really easy or should the definition be different? Is there a counterexample? That is, is there a bijective map $f: A \rightarrow B$ between two lattices/posets $A$ and $B$ such that $a \leq b$ implies $f(b) \leq f(a)$, but $f^{-1}$ does not have this property?