Are all morphisms in category theory are actually some kind of mappings? What is the importance of associativity rule in category theory?

Question

I always found the identity and the composition rule trivial while thinking about categories. But I think the associativity rule is NOT trivial and it is hard to create a composition table that obeys the associativity rule. Do you think there is something about the associativity rule that makes it very special and useful in mathematics, or am I just confused? I know we can create an associative composition table using mappings between sets and using permutations. So can we say all morphisms in category theory are actually some kind of mappings?

For example, this is a poset with 3 objects. And the morphisms are not mappings, they represent some relation between these 3 objects.

But I could turn the objects into specific sets, and turn the morphisms into specific mappings between those sets. So that the composition table stays the same and arrows are still associative. (I don't have to add every possible mapping as arrows, I just added one of them.)

So is it possible to turn every category into a combination of some sets and mappings between those sets?

Note: I am very new to category theory and abstract algebra, sorry if this question is confusing.

Answer 1

Thanks to @Berci, I found out that every small category is isomorphic to some subcategory of Set. This is explained by Cayley's Theorem and Yoneda Lemma, which I now understand are very fundamental subjects of category theory. This means every object in a small category can be represented as a set, and every morphism can be represented as a mapping.

Cayley's Theorem: https://proofwiki.org/wiki/Cayley%27s_Theorem_(Category_Theory)#Theorem

Yoneda Lemma: https://en.wikipedia.org/wiki/Yoneda_lemma

Thanks to everybody who answered my question. I can edit my answer if I got something wrong. I will try to learn more about Cayley's Theorem, Yoneda Lemma, large categories, and why we need them.

Answer 2

(1) Yes associativity is extremely useful, especially when doing algebra (e.g. module theory or commutative algebra), e.g. 2 fold associativity gives us the generalized (finite-fold) associativity in a category (just by induction)

(2) No morphisms need not be mappings, often times they are equipped with a few mappings, or even an indexed family of mappings (if by mappings you mean set maps), there are categories like ones given by monoids and groups where the morphisms are not really mappings at all, they are elements of the monoid or group (resp.) in this case for instance.