Functor to the empty (objectless) category? My textbook says:

Let $G$ and $H$ be monoids, regarded as one-object categories $A$ and $B$ respectively. A functor $F: A \mapsto B$ must send the unique object of $A$ to the unique object of $B$, so it is determined by its effect on maps.

The thing I can't understand is why $F$ must act in such a way? As far as I know, there exists the empty category with no objects at all, so one has freedom to define $F': A \mapsto C$ where $C$ is abovementioned objectless category.
Am I wrong?
P.S.
Seems like I try to interpret empty category $C$ as a fancy kind of monoid/one-object category, which has (logical surprise!) nothing inside and hence every statement about its internal structure is vacuously true. Not quite understand whether I am allowed to think in such a way or not.
 A: There is indeed a category $E$ with no objects (and therefore no morphisms). However, if a category $A$ has at least one object then there does not exist any functor $A\to E$.
Let's think about sets for a moment:
How many maps of sets are there from a non-empty set $A$ to the empty set $ \emptyset$? None! Because a map $A\to \emptyset$ needs to assign to every element $a\in A$ (of which there is at least one) an element in $\emptyset$; but there are no elements in $\emptyset$.
How many maps are there from a $1$-element set $\{a\}$ to another $1$-element set $\{b\}$? One! Namely the one given by $a\mapsto b$.
It is the same for functors:
A functor $A\to E$ from a non-empty category to the empty category needs to assign to every object $a$ in $A$ (of which there is at least one) an object in $E$; but there are no such objects!
Let $A$ and $B$ be two categories with exactly one object $a$ and $b$ respectively and let $F\colon A\to B$ be a functor. Then $F$ must send $a$ to an object $F(a)$ in $B$. But there is only one object in $B$, namely $b$, hence $F(a)=b$.
