There are some theorems proven from diagram chasing. However, all the proofs I saw was proving those statements when objects are all sets. In this case, this category is locally small, but not all locally small categories have objects as sets. So I think "locally small" category should be named for subcategories of Set.
For example, when proving snake lemma, we do a digram chasing. However, when proving this theorem, we pick elements and chase the diagram. That is, we prove it by assuming that all objects are sets and $Hom(A,B)$ is a subset of $B^A$. Let call this condition, the condition-@. (Note that this condition is even stronger than small category)
In an arbitrary category, when $f:A\rightarrow B$ is an epimorphism, we cannot simply pick an element of $B$ and choose and element of $A$ that is mapped to the element of $B$ under $f$. Indeed, the phrase "element of $B$" is nonsense since we don't know what $B$ is. So I do not get why we need locally small categories. It seems too general that most arguments cannot be applied.
Secondly, it is said everywhere that Snake lemma and Five lemma and etc can be proven in any abelian category, but I don't get how the usual diagram chasing argument can be applied to prove those when a given category does not satisfy the condition-@.
Of course, the snake lemma holds for the category of chain complexes of $R$-Mods, namely $Ch(R-Mod)$, but this category does not satisfy the condition-@, since $Hom(C,D)$ is no more a subset of $D^C$. However, the same idea of diagram chasing can be applied to this category to prove the snake lemma.
So is there a condition that generalizes the condition-@, so that we can apply diagram chasing argument to all kinds of $Ch(Ch(....Ch(R-Mod)...))$?