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Suppose that $f: \emptyset \to \emptyset$ is the empty function. The only possible inverse function, I believe, is that for which there are no elements in the domain or codomain, and the statement $(f \circ f^{-1})(x) = x$ for all $x$, and vice versa, holds vacuously.

Is this correct?

user861776
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    If $f\colon A\to B$ is any bijective function, then the domain of $f^{-1}$ is always $B$ and codomain of $f^{-1}$ is always $A$. Since in this case $A=B=\emptyset$, then certainly $f^{-1}$ is a function from $B=\emptyset$ to $A=\emptyset$. – Greg Martin Mar 07 '21 at 18:46

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