On the notion of (locally) small categories I've read many related questions like this and this, but I am still confused about what (locally) small categories are. My text is "Category Theory Lecture Notes" by Turi in which one reads

A category $\mathbb{C}$ is small if its collection of objects $\text{Obj}_{\mathbb{C}}$ and
collection of arrows $\text{Arr}_{\mathbb{C}}$ are sets; it is locally small if the collection
$\mathbb{C}(A,B)$ of arrows from $A$ to $B$ is a set for each pair of
objects $A$ and $B$.

The text says the category of natural numbers $\boldsymbol{N}$ is small, and the category of finite sets $\boldsymbol{Finset}$ is locally small.
Let me elaborate on what I don't understand:

*

*The condition of being small is pretty loose to me, say, aren't the collections of arrows and objects of a category always sets? If not, can you please show a simple commutative diagram (of a category) in which this is not the case?


*Similarly for locally small categories, how can the collection $\mathbb{C}(A,B)$ not be a set? Any minimal example here is also appreciated.
 A: 
aren't the collections of arrows and objects of a category always sets?

No. We want to study many examples of categories given by all mathematical objects of a certain type, e.g. all sets, all groups, all vector spaces, all topological spaces, that kind of thing. The collection of objects in each of these categories are proper classes, meaning they are "too large to be a set."

Similarly for locally small categories, how can the collection $\mathbb{C}(A,B)$ not be a set? Any minimal example here is also appreciated.

There aren't any really "natural" examples here; essentially any category that arises in practice is locally small. It's standard to use "category" to mean "locally small category" without any further elaboration because it is a very mild and natural restriction.
However, if $C, D$ are two (locally small) categories, the functor "category" $[C, D]$, of all functors and all natural transformations, can fail to be locally small; Theo Johnson-Freyd gives an example here. This is not a big deal in practice because we rarely need to consider the entire functor category unless $C$ is small, and in that case there are no issues.
These "size issues" in category theory can mostly be ignored a lot of the time but they are important. You can consult Mike Shulman's Set theory for category theory for more details.
