How does a header file point to a library?

Question

C++ / C#

I started to get acquainted with make and asked myself this question.
For example, there is a connected header: In it, as I understand it, there are function prototypes, and the functions themselves are compiled into object files and compiled into a library, for example ".a". The linker sees the header line and goes to /usr/lib . But how does he see the right library? Goes through everything? #include<stdio.h>