I'm doing some independent study on Profinite Groups this summer and, as I understand it, it is important to be familiar with the notion of an inverse limit before doing so. The trouble for me is that there seem to be multiple approaches to teaching them, each depending on a different subject that sends me down a rabbit hole. At first, I was told it would be useful to approach it from the standpoint of Category Theory––looking at limits and colimits––but another, seemingly less general approach seems to be by looking at inverse systems of groups. (I think that an inverse system of groups exists within the category $\textbf{Grp}$, so this may be an equivalent definition... Apologies if this is wrong.)
How would you recommend learning about inverse limits for the purpose of studying Profinite Groups and, eventually, the Profinite Topology? Specifically, could anyone please recommend some sources on learning the subject? In case it is asked: of course I have googled around, but there seem to be so many approaches and variety of names (e.g. limits, colimits, direct limits, projective limits, inverse limits, etc...) that it's a bit overwhelming. If it's of any use, I've taken Calculus (including Multivariable), Analysis, Set Theory, Algebra I (Group Theory), and Algebra II (ring, field, and Galois theory). I also know some topology.
This may be more than you need, but you can find an introduction to categories and universal algebra, including the concepts of limit and colimit, in Chapters 1 and 2 of Jacobson's Basic Algebra II.
For an even briefer treatment, you could refer to the appendix on category theory in An Introduction to Homological Algebra by Weibel.