Trying to understand the very basics of category theory... While trying to solve a simple exercise on the subject I suddenly realized that I don't even know what morphism equality is! (The exercise asked to prove that such-and-such morphism composition in such-and-such category is well-defined - but to prove this I had to prove that it is associative, which requires me to prove equality of the sort $f;(g;h)=(f;g);h$ - which gave raise to this question)
On the one hand, many proofs in category theory consist of diagrams of some sort where it is shown that $f;g=h;j$ because a square whose edges are morphisms $f:A\rightarrow B$, $g:B\rightarrow D$, $h:A\rightarrow C$ and $j:C\rightarrow D$ 'commutes':
So it would seem that two morphisms are equal iff they lead from the same source to the same destination?
On the other hand, this would mean that there may be at most one morphism in one direction between two given objects - but this seems not the case.
I looked at Wikipedia article but could not find morphism equality defined.
Sorry if this is an extremely basic question - but when are two morphisms equal?
They are equal when they are the same morphism. There can be many morphisms from $A$ to $D$. One of them is $g\circ f$, and one of them is $j\circ h$. If they are the same morphism then they are equal, and if they are not the same morphism then they are not equal.
Category theory is meant to be an abstraction, so we can't say much more than that in general. However, in specific cases we can look a bit closer at what this means. For instance, in most concrete categories people encounter (sets, topological spaces, metric spaces, groups, rings, modules, etc.), morphisms are all just functions. So when are two functions equal? Well, that is a question I think you know the answer to: when $g\circ f(a)=j\circ h(a)$ for every $a\in A$.