# Why is every category in which all (even large) limits and colimits exist thin?

The wiki page on complete category states:

The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.

Why is that? Can we interpret any object in such a category as a limit candidate of some large diagram?

(Also, isn’t this little fact a nice motivation to really care about the distinction between sets and classes in category theory? Often enough I encountered the attitude “Yeah, we don’t care about classes or sets, just think of all those things as collections” – should I discard that attitude?)

Let $$C$$ be a category with all (large) limits. Assume that there are two morphisms $$f,g : a \to b$$, $$f \neq g$$. If $$J$$ is a class (or large set), then there are at least $$2^J$$ morphisms $$a \to b^J$$, where $$b^J := \prod_{j \in J} b$$. Applying this to $$J:=\mathsf{Mor}(C)$$, we obtain an injection $$2^J \hookrightarrow J$$, which is a contradiction.