This question was originally asked by Paul Slevin, but it was deleted before I had the chance to answer. It's actually quite subtle, so I thought it would be worth reposting it here.
Consider the following proposition: $\DeclareMathOperator{\mor}{mor}$
Proposition. Let $\mathcal{C}$ be a category. If $\mathcal{C}$ has limits indexed by diagrams of size $\left| \mor \mathcal{C} \right|$, then $\mathcal{C}$ must be a preorder.
This is easy to prove when $\mathcal{C}$ is a small category: assume, for a contradiction, that there are two distinct parallel arrows $f, g : A \to B$ in $\mathcal{C}$. Since $\mathcal{C}$ has large products, there exists an object $C$ in $\mathcal{C}$ equipped with a bijection $\textbf{Set}(\mor \mathcal{C}, \mathcal{C}(X, B)) \cong \mathcal{C}(X, C)$ that is natural in $X$. In particular, we have $\textbf{Set}(\mor \mathcal{C}, \mathcal{C}(A, B)) \cong \mathcal{C}(A, C) \subseteq \mor \mathcal{C}$ but the LHS has cardinality $\ge 2^{\left| \mor \mathcal{C} \right|}$, so this contradict's Cantor's theorem on powersets.
Question. How do I adapt this proof for the case where $\mor \mathcal{C}$ is a proper class? In class–set theories such as von Neumann–Bernays–Gödel or Morse–Kelley, it doesn't make sense to talk about the collection of all subclasses of a proper class, because any member of any class is a set, and even the collection of all maps $\mor \mathcal{C} \to \mathcal{C}(X, B)$ fails to be a class.