For example, in set theory, all injective maps ($f:X\rightarrow Y$) (monomorphisms in the category of sets) admit a left inverse ($g:Y\rightarrow X$ such that $g \circ f = id_X$). This is not true in a general category. That is, the morphisms with this property in a general category aren't only monomorphisms, they're split monomorphisms, so set theoretic intuition absolutely fails here. There are countless other examples, but I'm sure you see my point.