How do you know two morphisms are equal (without using elements) Given two morphisms in some category, which is to say that you are told that $f$ and $g$ are in the cat $C$ and nothing more, how can you know if they are equal?  Normally we appeal to the elements of their domain and codomain.    Suppose you can't say anything about these sets, like you know nothing about the elements of these sets and hence how the functions act on them.  How much data about $C$ must you have to say that these morphisms are equal?  What data do you need?
 A: I assume you are interested in general techniques to prove when two morphisms are equal/different.
In general it is easier to prove when two morphisms are different. For instance, if the morphisms have different sources or targets they are different. Another way to distinguish two morphisms is by testing them against other morphisms: if $f,g \colon A \to B$ are two morphisms, if there is a $h \colon X \to A$ such that $f \circ h \ne g \circ h$ then clearly $f \ne g$ (a similar argument applies to morphisms of the form $h \colon B \to X$).
Generally, if you want to prove that $f$ and $g$, as above, are equal you should search for a special morphism defined either over $A$ or $B$ and test it (i.e. compose it) with both $f$ and $g$ to prove equality.
For instance if $h \colon X \to A$ is an epi-morphism and you are able to prove that $f \circ h=g \circ h$ then it follows that $f=g$. Similarly if you have an $h \colon B \to Y$ which is a mono-morphism and prove that $h\circ f=h \circ g$ then it would follow that $f=g$.
Another example could be the following: assume you know that $B$ is a sub-object of some product, so you have a family of objects $(B_i')_i$, a product $(B',(\pi_i \colon B' \to B_i'))$ and a mono-morphism $e \colon B \to B'$, then you have that the family $\pi_i\circ e \colon B \to B_i'$ is jointly-monic and so if you prove that for each $i$ the equality 
$$\pi_i \circ e\circ f = \pi_i \circ e \circ g$$
holds it follows that $f=g$.
As you can see, in all these cases we have used some morphisms defined over the source $A$ (the epi-morphism case) or the target $B$ (the projections above) and tested (composed) them with the two morphisms we wanted to prove(disprove) equality.
You can find many other example of this sort and I would even dare to say that probably every way for comparing morphisms in a category should fall in this very general method.
Hope this helps.
A: One answer is separating objects.
Say an object S is separating for  iff
for any morphisms f, g : A ⟶ B in , we have
$$(∀ x : S ⟶ A • f x = g x) ⇔ f = g$$
(where the bullet • divides the declaration and body of the quantifier.)
That is, morphisms can be distinguished by looking at their behaviour with S-generalised
elements.
Then one says S₁, …, Sₙ are separating iff
$(∀ i • ∀ x : Sᵢ ⟶ A • f x = g x) ⇔ f = g$.
Examples
In e,  ≔ { ⋆ } is separating and this fact is usually called `extensionality'.
In , the category of graphs and graph morphisms,
the naked dot  ≔ (graph with one dot) and arrow  ≔ (graph with two dots s,t and edge a : s ⟶ t) are separating.
Exercise :: what categories are separating for , the category of
(small) categories.
Hope this helps :-)
