I'm studying limits and colimits and more precisely I'm looking at forgetful functors and I'm trying to see if they preserve limits and colimits. In order to do that I first look at terminal and initial object and check if they are preserved, if not then I can conclude, but if they are preserved I don't know what else I can check.
For specific cases I try to build a left or right adjoint, but I was wondering if there was a more universal method? Like does the fact that terminal or initial object are preserved tells us something about limits and colimits in general? Because it seems that this might be true with the examples I looked at but I don't know if this is true in general.
A functor preserves all small limits if and only if it preserves equalizers and all small products (i.e. products of an arbitrary set-indexed family of objects). Dually, a functor preserves all small colimits if and only if it preserves coequalizers and all small coproducts.
Just preserving the terminal object (the empty product) is not sufficient to check that a functor preserves limits.