As per my understanding, C libraries must be distributed along with compilers. For example, GCC must be distributing it's own C library and Forte must be distributing it's own C library. Is my understanding correct?
But, can a user library compiled with GCC work with Forte C library? If both the C libraries are present in a system, which one will get invoked during run time?
Also, if an application is linking to multiple libraries some compiled with GCC and some with Forte, will libraries compiled with GCC automatically link to the GCC C library and will it behave likewise for Forte.
GCC comes with libgcc which includes helper functions to do things like long division (or even simpler things like multiplication on CPUs with no multiply instruction). It does not require a specific libc implementation. FreeBSD uses a BSD derived one, glibc is very popular on Linux and there are special ones for embedded systems like avr-libc.
Systems can have many libraries installed (libc and other) and the rules for selecting them vary by OS. If you link statically it's entirely determined at compile time. If you link dynamically there are versioning and path rules which come into play. Generally you cannot mix and match at runtime because of bits of the library (from headers) that got compiled into the executable.
The compile products of two compilers should be compatible if they both follow the ABI for the platform. That's the purpose of defining specific register and calling conventions.