I'm going through an algebra course that starts with category theory, and the definition of a category given is actually that of a locally small category, i.e. we assume that $\text{Mor}_{\mathcal{C}}(X,Y)$ is a set.
The issue arises when we start considering a category that has as its objects functors between two categories, and as its morphisms the natural transformations between said functors. The author said that this is "morally" not a category (hinting at 2-categories). I don't really understand what is meant by "morally" not a category. Is the only problem that the morphisms form a class? If so, why are we more interested in locally small categories specifically? It seems like allowing the objects to be a class is fine, but having too many morphisms could be a problem. Is it because morphisms are "more important" than the objects themselves (following Grothendieck)?
More generally, what are striking differences between large and locally small categories?
This is a duplicate question, but I cannot find the duplicate right now.
It has been already explained why locally small categories are important (tons of examples, and the Yoneda Lemma). Examples of categories which are not locally small have been given at SE/219539 and MO/3278. Here is an attempt to answer the set-theoretic concerns.
First of all, Grothendieck universes are very useful to manage size-isszes in category theory. For example, locally small categories can be considered to be small in this way. In detail:
Let $U$ be a Grothendieck universe. A $U$-small (or just: small) category is a category such that both morphisms and objects form sets in $U$, also called $U$-small sets.
Many categories in practice are not small, but they are locally small. But in order to formalize this, it is best to assume another Grothendieck universe $U \in V$. Then we define* a locally small category to be a $V$-small category such that all hom-sets are $U$-small. Notice that such a category is still small, just with respect to a larger Grothendieck universe. This is very useful, since you can apply all of the known constructions of small categories also to all categories, when you keep track that you are working with $V$-small categories. In particular, we can consider functor categories for $V$-small categories. So from this perspective, there are no "large categories".
If $\mathbf{Set}$ denotes the category of $U$-small sets, for every locally small category $\mathcal{C}$ we have the Yoneda embedding $\mathcal{C} \to \mathrm{Hom}(\mathcal{C}^{\mathrm{op}},\mathbf{Set})$, $A \mapsto \mathrm{Hom}(-,A)$. Notice that both sides here are $V$-small. The Yoneda embedding is very important, since it reduces many claims from an arbitrary locally small category to the category of sets.
*Unfortunately, this is not the only definition in the literature. See MO/3409.