Define "equality"in category theory

When I was trying to solve problem for 67D, I had the following though "what the hack does it mean that two morphism are different?". I mean when the maps are functions, we know what "equal" means, but it is not obvious when dealing with category. I know giving definitions might not be possible without adding conditions(I guess that's the reason of epic and monic concepts). The intuition that I had when trying to solve 67D was to think that the maps being different is equivalent to "exists $k$ such that $k\circ f \ne k\circ g$ or $f\circ k \ne g \circ k"(I know that this is maybe incorrect, but this could be maybe added as an intuition which helps to understand what morphism and objects are). Anyways, feel free to ignore this.