Given a category $C$, the pro-objects of $C$ are defined as cofiltered diagrams of objects in $C$. However, morphisms are not defined as natural transformations between diagrams but by some other procedure.
I guess the idea is that we are thinking of the diagram in terms of its limit. So I believed that if cofiltered limits exist in a naturally enlarged category, then the morphisms between pro-objects are merely morphisms between the limits of the diagrams. I am wrong and the n-lab page on profinite groups (Remark 1.2) indicates that
In most cases, the limit would not actually exist in the category of finite groups, and while it would exist in the category of all groups, it would be “wrong” category-theoretically: maps between profinite groups are not the same as maps between their honest limits in Grp.
Now, I am really confused about the maps in a pro-object category. How should I imagine them? What is the distinction between the maps between limits of objects and maps between pro-objects (say in Sets, Groups...)?