Let $X$ be some space, $A$ a subspace of $X$, $f:X \rightarrow X$ a function. My first guess for what $f|A$ is would be that the domain of $f$ is restricted to $A$, but I can't find any confirmation that this is actually the case. The only notation that I'm aware of is $f|_A$.
For reference, the actual place that this question arose in is in Hatcher's Algebraic Topology page 2:
A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t:X \rightarrow X$, $t \in I$, such that $f_0 = \mathbf{1}$ (the identity map), $f_1(X) = A$, and $f_t|A = \mathbf{1}$ for all $t$.
You are correct, this is indeed just a restriction.