Prove the forgetful functor $U: Grp \to Set$ preserves limits

Question

Let $F: A \to B$ be a functor. F preserves limits iff any for any diagram $ D: I \to A$ s.t. $\operatorname{lim}_I D $ exists, $\operatorname{lim}_I FD$ exists and $\operatorname{lim}_I FD \cong F( \operatorname{lim}_I D )$.

I'm having difficulty showing that the forgetful functor $U : Grp \to Set$ preserves limits. Using the definition doesn't seem to get me anywhere, as it seems I just don't have enough information about the limit. Am I missing something or is there a better way to go about this directly ( avoiding adjoint functor theorem etc.)?

Answer 1

It can be shown that limits in general can be constructed from products and equalizers (see Awodey's "Category Theory" 5.21 and 5.22 for example). Using this it suffices to check that the functor preserves products and equalizers, which is clear.

Answer 2

If you’re aware that hom preserves limits in its right argument, then you might like the approach of showing that $$U\ \simeq\ \text{Hom}_{\textbf{Grp}}\left(\mathbb{Z},-\right)$$ and then deducing the result automatically.