Given a partial function $f : A \rightarrow B$, and a subset $S \subseteq A$, we get a new partial function $$f \restriction_S : A \rightarrow B$$ by restriction. However, I prefer to analyse $f \restriction_A$ as a composite. To each subset $S$ of $A$, there's a corresponding a partial function $[S] : A \rightarrow A$ satisfying $\mathrm{supp}[S] = S$ and $S \subseteq \mathrm{id}_A$, that could be called the "restrictor" for $S$ or something like that, because for each partial function $f : A \rightarrow B$, we have: $f \restriction_S = f \circ [S].$ I much prefer the latter notation; for starters, we can put the restrictor on the other side to restrict at the codomain, as in $[S] \circ f$. It also makes it really obvious that $g \circ (f \restriction_S) = (g \circ f) \restriction_S,$ which just ends up being a special case of associativity. Also, to some mathematicians, the $\restriction$ notation indicates a change of domain, but in the partial function viewpoint, we usually don't wish to change domain, it's still just $A$.

Anyway, it would be nice to have standard notation for what I'm denoting $[S]$. Is there one?