What are examples of properties of particular categories that can be formulated in categorical language and "feel" like they ought to be provable formally but they actually are not? I think that this question is inherently imprecise but one can not seriously say that every question on this site is precise. 

To clarify, the existence of fiber products in the category of schemes would not count IMHO because I personally do not expect a random category to have fiber products.

A **non**-example is the fact that for a morphism of schemes $X\rightarrow Y$, the diagonal morphism $X\rightarrow X\times\_Y X$ is a monomorphism. At first I thought that this could only proven non-formally by noticing that the diagonal morphism is an immersion and immersions are monomorphisms but [it turns out](https://math.stackexchange.com/questions/87042/is-the-diagonal-morphism-always-a-monomorphism) that in any category with fiber products the diagonal morphism is a monomorphism.