What does 'equal' mean for morphisms of a category?

Question

I don't quite understand what does equal mean for morphisms in category theory. Since object of a category is not necessarily a collection of some elements, then how to decide whether two morphisms are the same or not? Is it allowed that for some category $C$, there exist two different morphisms $f,g$, but for any possible h which is not a identity morphism, $f\circ h = g\circ h$ and $h\circ f = h\circ g$ (so that there does not exist any monic or epi morphisms in this category, and any objects $a$ of this category do not have any structure such as $x \in a$)? Or we can never know it?