In the book Mathematics Made Difficult; at the top of page 12 the author states
Further, assume initial and final objects $0,1$ and a contravariant functor from the category to itself $\cdot'$ that is alternating on objects and preserves no arrow except $0 \to 1$.
What does it mean for a functor to alternate on objects? I searched up 'alternating functor' but I can't seem to find any sources for this term.
I'm pretty sure "Alternating" here just means that it is an involution. That is, for any object $A$, $(A')'=A$. This is not a standard terminology that I know of, but it makes some intuitive sense and is the only reasonable interpretation I can see that makes sense in context. (In particular, what is being asserted here is that if $L$ is a distributive lattice and $\cdot':L\to L$ is an "alternating" order-reversing map such that $A\leq A'$ and $(A')'=A$ implies $A=0$, then $L$ is a Boolean algebra and $\cdot'$ is the complement map. If "alternating" means $(A')'=A$ for all $A$, this is true and is a cute exercise to prove.)