I have read (I can't remember where) that, where $V$ is some set, the notation $\mathcal{P}(V) \nrightarrow \mathcal{P}(V)$ is used in category theory to denote a binary relation $R \subseteq \mathcal{P}(U)\times \mathcal{P}(U)$.
Is this correct? If so, what is the definition of the symbol $\nrightarrow$ in category theory?
I have also seen as examples of such notation (in notes I took in some talk)
$$ A \stackrel{\text { some }}{\nrightarrow} X \Longleftrightarrow X \cap A \neq \emptyset $$
to symbolise the existential quantifier defined on some universe of discourse $X$, and where $A \subseteq X$. Is this abuse of the symbol ${\nrightarrow}$?