Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle then no locally presentable category contains a full subcategory which is large (= a proper class) and discrete (= contains no nonidentity morphisms).
On the other hand, I know that many non-set theorists find the constructible universe $V = L$ to be a very appealing place to do mathematics, for a variety of philosophical reasons.
Now, my understanding is that $V =$ a Vopěnka cardinal is a very Large cardinal axiom, and in particular is inconsistent with $V = L$.
So what I would like is an explicit example of a category $C$ which is locally presentable for $V = L$ and a full discrete subcategory of $C$ which is large for $V = L$.
This is very closely related to this previous MO question: Can Vopenka's principle be violated definably?
The difference is that there the OP was asking for a single definable class which violated VP for any universe where VP fails. Here I am asking if such an example can be given in the special case of the constructible universe.