I have the following problem
Describe the finite limits and the filtered colimits in the category $Grp\ $ and show that they commute.
This is my first "more practical" exercise on calculating limits and colimits. I have some theoretical background but I couldn't do this. I know that on taking the colimit you have to take the coproduct and make a quotient by some equivalent relation (but I kind have the case for $Set$ and it isn't quite clear how I can apply this in $Grp$)... and basically the same for limits.
On showing that they commute, I was wondering to use the fact that they commute in the category of $L_{Grp}$ structures and then prove that $Grp \hookrightarrow L_{Grp}$-$Str$ is closed under taking finite limits and filtered colimits. But I couldn't do this specially by not knowing the first question. But I also have an example that creates a "canonical" multiplication on $\sqcup _{i\in I} F(i)/\sim$ and prove that this has a structure of group, but is this the colimit in $Grp$? I know that it is on $Set$, but coproducts in the group category isn't the disjoint union ... so I got stuck.
I would appreciate some detailed hints due the fact that this is my first "practical" exercise.