I have read that a product of objects $\{A_i\}_{i\in I}$ in a category $\mathcal{A}$ can be defined as a limit of a discrete diagram i.e. a diagram $D:\mathcal{I}\to\mathcal{A}$ where the only morphisms in $\mathcal{I}$ are the identity morphisms.
So if we define $\mathcal{I}$ to be the discrete category whose objects are the elements of the set $I$, and where $D(i):=A_i$ for each $i\in I$, then the limit of this diagram will be $\prod_{i\in I}A_i$ with the projections $\pi_i:\prod_{j\in I}A_j\to A_i$.
However, consider the following: Let $D$ be a discrete diagram in Set that has a limit and where the objects form a proper class. Then the limit is a product $\prod_i D_i$ indexed over a proper class, rather than a set. My understanding of taking arbitrary products of sets is that it must be indexed by a set, yet the definition here seems to include proper classes. Am I missing something?
EDIT: In other words, a product of sets involves using an index set, whereas the category theory definition of a product seems to allow for indexing by proper classes, unless we restrict to discrete diagrams that are also small.
Best Answer
The category of sets has all small limits, where small means that the diagram $D$ is a small category. In particular $D$ has a set of objects, so the notion of "limit of small discrete category $D$" agrees with the notion of "product of a set-indexed family of sets".
Most large diagrams in Set just don't have limits. In fact, if a category has all products, by which I mean limits of all discrete diagrams, not just small ones, then it is a preorder! (Meaning there is at most one arrow between any two objects.) See Theorem 2.1 of Mike Shulman's Set Theory for Category Theory for the quick argument, and lots of other enlightening discussion.
As Eric Wofsey notes in the comments, usually people only talk about small diagrams. In particular, the terms "complete category" and "cocomplete category" mean that the category has all small limits and colimits, respectively.