I am reading "Set Theory and General Topology" by Takeshi SAITO (in Japanese).
This book contains the following problem:
Let $X$ be a set.
Let $F, G : \text{Map}(X, X)\times\text{Map}(X, X)\to\text{Map}(X, X)$ be mappings such that $F(f, g) = f\circ g$ and $G(f, g) = g\circ f$.
What is the necessary and sufficient condition on $X$ that ensures $F=G$?
The author's answer for this problem is here:
The necessary and sufficient condition on $X$ that ensures $F=G$ is $|X|\leq 1$.
If the number of elements of $X$ is less than or equal to $1$, then $\text{Map}(X, X)=\{\text{id}_X\}$.
So, $F=G$.
Let $a, b\in X$ and $a\neq b$.
Let $a, b : X\to X$ be constant mappings such that $a(x) = a$ and $b(x) = b$ for any $x\in X$.
Then, $F(a, b) = a\circ b = a \neq b=b\circ a=G(a,b)$.
So, $F\neq G$.
Let $X=\emptyset$.
In this case, I think we cannot compose two empty mappings, so we cannot define $F$ and $G$.
So I think the necessary and sufficient condition on $X$ that ensures $F=G$ is $|X|= 1$.
Am I wrong?
Sure you can compose two empty mappings. Let $e$ be the empty map. Then $e\circ e$ is, by definition, the function which maps all elements $x\in\emptyset$ to $e(e(x))$. Since there are no $x\in\emptyset$ in the first place, the empty map does this vacuously: it maps all of the necessary $x$ (explicitly: none are necessary) in the prescribed way.