TITLE: In the morphisms-are-functions view of category theory, how are poset categories explained?
QUESTION [5 upvotes]: Some people have taken the view that morphisms in category theory ought to simply be called functions.
However, not all morphisms look like functions at first sight. For example, in categories of partial orders, such as the category in which the objects are the natural numbers and a morphism exists between two numbers x and y if and only if x < y: How is, say, the morphism that corresponds to the fact "2 < 3" viewed as a function? What is its domain and codomain?

REPLY [4 votes]: Let me offer another point of view.
I do not think that invoking Cayley's theorem may justify calling morphisms functions, because such a representation generally does not give a full subcategory of sets, and non-full subcategories rarely inherit interesting properties from they embedding category. It would be better perhaps to invoke the Yoneda lemma, which says that a locally small category $\mathbb{C}$ fully embeds into the category of presheaves $\mathbf{Set}^{\mathbb{C}^{op}}$. A presheaf $F \colon \mathbb{C}^{op} \rightarrow \mathbf{Set}$ is essentially a (somehow generalised) $\mathbb{C}$-sorted algebra, where $\mathbb{C}$ may be though as a generalised "signature together with equations". A morphism of such algebras $h \colon F \rightarrow G$ consists (as usual) of a family of functions (so yes, technically, it is not a function, but a family of functions, but it is easy to represent such a family as a single function) between carriers $h_C \colon F(C) \rightarrow G(C)$ compatible with the operations taken from the signature. This means, that every category can be realized as a full subcategory of generalized algebras and algebra homomorphisms (BTW: the Cayley's theorem is obtained form the above by "gluing" carriers of the algebras).
Nonetheless, I do not think invoking the Yoneda lemma is a good idea either.
What I think, is that one should first ask what a function is. And because we are interested in mathematical foundations, we cannot just say that a function is a subset of a Cartesian product of two sets satisfying some properties, without saying what our sets are, or more accurately, what set theory are we willing to work in. Accidentally, it seems to be a difficult task to give one right answer. Some may say that a function is something that lives inside ZF set theory, or any consistent extension of ZF. However, there are other theories that we are willing to accept as “set theories” which are not in any sense extensions of ZF.
Categorically, one could say that a "set theory" is an elementary topos. However, again, there are “set theories” without the full power-set axioms, which constitute some weaker systems (like $\Pi$($W$)-pretoposes). Moreover, in computer science, the concept of a function is even more general --- a function is no longer though as a set of mere points linking arguments with results, but as a kind of a computational process which on given input produces a result. Such functions are defined in various lambda calculi. For example, simply typed lambda calculi are tantamount to Cartesian closed categories, and untyped lambda calculi correspond to categorical Lambek’s C-monoids.
What I would like to argue is that to speak about functions we do need far less structure then we have in a "set theory", in a "topos", and even less then we have in a "Cartesian closed category" --- in some sense the concept of a category is the best attempt to give a purely abstract axiomatization for the concept of function!

Let me also point that it is crucial to distinguish the concepts from our meta-theory, from the internal concept that we have to define inside the theory. It is clear, that Mariano’s table (see the comments to the original question) is not an external function, but it does not mean, that it cannot be though as an internal function in a category --- similarly the fact that there exists a model of ZFC build upon natural numbers, does not imply that functions in ZFC are natural numbers (and not functions!).