$\DeclareMathOperator{\colim}{colim}$ I've been having trouble making sense of various notational / conventional aspects of limits.
The first introduction I had with them was with a set of lectures notes by Schapira, where he defines the colimit of a functor $F: I\to C$ and the limit of a functor $G:I^{op}\to C$. The way he introduces limits/colimits is as an object representing a functor $I^{op}\to Set$. So first he defines what a projective limit (of a functor $I^{op}\to Set$) is in $Set$ and then (keeping the notations above) takes $\colim F$ and $\lim G$ to be (if they exists), objects representing the functors $X\mapsto Hom_{C}(F(-),X)$ and $X\mapsto Hom_{C}(X,G(-))$. From this I sort of internalised that you take the colimit of a functor $I\to C$ and the limit of a functor $I^{op}\to C$, but that the limit of $I\to C$ wouldn't make sense.
But when you look at the definition with universal cones/cocones, say from Riehl's book, nothing prevents me from taking the limit / colimit of the same functor, simply by looking at cones under or over the given functor, and I've read examples where one actually does that. But with my original definition, it's not given that you can take the limit of $F:I\to C$, only the colimit. Even simply when thinking of functors as diagrams, it's obvious you should be able to have both a limit and colimit of the same diagram, but I'm not sure how to deal with that with the first definition. I'm pretty sure Schapira's right with his definition and I just don't understand how to reconcile both.Maybe this stems from the fact that $Fun(C,D)^{op}\simeq Fun(C^{op},D^{op})$, I'm not sure.