I'm currently working through Rotman's "Introduction to Homological Algebra" and as I'm going through chapter 5 I'm noticing how he's defining direct limits, projective limits, inverse systems in the book but so far I haven't found any Category Theory books that utilize his level of category theory. Right now I'm going through Steven Roman's text "An Introduction to the Language of Category Theory" and it doesn't cover category theory the way Rotman's book presents it.
As an example I haven't found any category theory books which don't show the diagrams with the $\underrightarrow{limM_i} $ and only Rotman's text illustrates it.
I'm not sure if this is only practiced in homological algebra or perhaps category or if it might only be rotman's style.
My goal is to get to Sheaf Theory which uses the $\underrightarrow{limM_i} $ notation alot and I find it necessary to see it applied in category theory aswell as Homological Algebra.
Bredon's Book "Sheaf Theory" for example uses this notation like its going out of style and it makes me want to see it more. Maybe more familiar examples are $\underrightarrow{lim} (F_i\oplus A) $ and $\theta: Hom_R(\underrightarrow{lim} M_i,B) \rightarrow \underleftarrow{lim}$ $Hom_R(M_i,B)$
Any advice and suggestions would be welcomed.
$\varinjlim$ (respectively $\varprojlim$) is nothing more than a colimit (resp. limit) whose shape category is a directed (resp. codirected) set.
Since category theory works extensively with the general case of limits and colimits of arbitrary shape, there is no need to introduce these as a special case.
It also turns out that (co)directed sets aren't quite the right notion: better is the slightly more general notion of a (co)filtered category.
Filtered colimits (and sometimes, the dual notion of cofiltered limits) are very important in category theory. You will surely be able to find them in your references now that you know to look them up under that name.