To be more specific, let $\mathsf{Grp}$ and $\mathsf{Set}$ denote the categories of groups and sets respectively. We know there exists a pair of adjoint functors $U\colon \mathsf{Grp}\to\mathsf{Set}$, the forgetful functor, and $F:\mathsf{Set}\to\mathsf{Grp}$ the free functor. We proved that $F\dashv U$ implies $U$ preserves limits and $F$ colimits (yes it holds in general for any given couple of adjoint pairs).
What bothers me is the following:
Limits in $\mathsf{Grp}$ are done as in $\mathsf{Set}$.
I really don't understand why, because I would expect it if we had a reflective functor. Point is that given a diagram $\mathbb{D}$ and its limit $(L,l_i)$ in $\mathsf{Grp}$, then being $U$ preserving we have $\mathcal{Lim}_{\mathsf{Set}}U\mathbb{D}=(UL,Ul_i)$ so it makes more sense to me to say the opposite of the previous sentence:
Limits in $\mathsf{Set}$ are done as in $\mathsf{Grp}$
Which isn't as much satisfactory of course. I hope I made clear what confuses me.
The relation "X is done as Y" is clearly symmetric. So it doesn't really matter here which formulation you pick. But, in context there is a difference, because usually you first construct limits in the category of sets, and then look at more complicated examples, perhaps starting with some familiar and basic categories such as the category of groups. Then you realize that it can be done in such a way that the underlying set is just the limit of the underlying sets, i.e. the forgetful functor preserves limits. In fact, something much stronger is true: the forgetful creates limits. And generally speaking, when we already know Y and are currently investigating X, one prefers to say "X is done as Y".
To be precise, of course the construction of limits of groups involves more than just writing down the underlying set, so actually it is more complicated than the construction of limits of sets. To unify them, however, we can construct limits of algebraic structures of any type (function symbols with arities and equations between them). Groups are algebraic structures with function symbols $e^{[0]},i^{[1]},m^{[2]}$ (the exponents denote the arity) and equations $m(x,e)=m(e,x)=x$, $m(x,i(x))=m(i(x),x)=e$, $m(x,m(y,z))=m(m(x,y),z)$, and sets are algebraic structures with an empty set of function symbols and an empty set of equations. Limits of all algebraic structures (sets, groups, lattices, vector spaces, etc.) are done the same way: we take the limit of the underlying sets and then define the operations pointwise.