What does $\Phi\vert_S$ mean?

Question

In Abstract Algebra (Dummit and Foote) Exercise 11 in Section 6.3 (A Word on Free Groups) it says:

... Prove that $A(S)$ has the following universal property: if $G$ is any abelian group and $\varphi:S\rightarrow G$ is any set map, then there is a unique homomorphism $\Phi:A(S)\rightarrow G$ such that $\Phi\vert_S=\varphi$. ...

The notation is not introduced in the book and I wouldn't like to take any wrong assumption.

Answer 1

One way of saying it is, if we consider the inclusion $i:S\hookrightarrow A(S)$, then $\Phi|_S=\Phi\circ i$.