Definition of Limit and Foundational problems. I am new to Category theory and I have a quite strong foundational problem.
For example, let's start from the definition of Limit stated by wikipedia (https://en.wikipedia.org/wiki/Limit_%28category_theory%29). A limit is an universal cone that satisfies the universal property, i.e. "for each....there exist..." where ALL the variables are not requested to be sets, and consequently they make no sense by a Set-Theoretic point of view.
The same happens every time I use an universal property, or also, for example, in the basic definition of Functor (for each objects...for each morphism...).
I am getting completely stuck and incapable to move on...how can this problem be solved?
 A: There are various foundations of set theory and mathematics which make category theory work. See Shulman's Set theory for category theory for an overview; in fact there is a huge amount of literature on this topic. If you really want to work with ZFC or NBG (which is certainly not the best choice!) then a functor $F : \mathcal{I} \to \mathcal{C}$ consists of two class functions $F_{\mathrm{Ob}} : \mathrm{Ob}(\mathcal{I}) \to \mathrm{Ob}(\mathcal{C})$ and $F_{\mathrm{Mor}} : \mathrm{Mor}(\mathcal{I}) \to \mathrm{Mor}(\mathcal{C})$ satisfying the usual laws. A class function is a class which satisfies the usual definition of a function. In the definition of a limit of a functor, we quantify over functors and therefore over classes. This is not possible in ZFC. But it seems to be possible in NBG.
Notice that you won't run into any set-theoretic difficulties if you don't work with limits of functors, but rather with limits of diagrams as defined here for instance. At least, in the definition of a limit, the index category should be assumed to be (essentially) small.
In the comments it has been said that all objects are sets. In ZFC we model our mathematical world that way. In particular, say, Euler's number $e$ is a set. Do we really want this? I think that the paradigm "everything is a set" is outdated since many decades. For more in this, see Leinster's Rethinking set theory (this is based on Lawvere's ETCS), Shulman's SEAR, Lawvere's ETCC, and Voevodsky's Homotopy type theory. See also MO/8731.
