Very basic questions about definitions in Category Theory I have read this question but I have a more basic question, I think.  I seem don't understand the very concept of a morphism.  
As morphisms are described they always seem to be a mapping from one object to another.  In other words, they are a function taking an object and returning another object.  This view is reinforced by the use of arrows in diagrams describing categories where any $f$ "transitions" from one object to another (similar to a state transition diagram).
However, when looking at a description of partial orders on the Haskell wiki this doesn't seem to be the case.  There they describe a partial order as a morphism between any two objects $A$ and $B$ iff $A \leq B$.  But there doesn't seem to be a 'transition' or mapping between the two.  Just a relationship ($\leq$).  At best this is a mapping (function) taking 2 objects and returning a boolean value.
So, what 'is' a morphism if it isn't a function?
 A: I remember having had a similar moment when I was first learning about categories. For me it was set off by the opposite category $\mathcal C^{op}$, which is defined by $Hom_{\mathcal C^{op}}(A,B)=Hom_{\mathcal C}(B,A)$ and $f^{op}\circ g^{op}=(g\circ f)^{op}$, that is, the composition is reversed.
This is one of the first examples of a really abstract category, a category in which the morphisms have nothing to do with a functional way to get from $A$ to $B$. If I have $f:G\to H$ in the opposite category of groups, $f$ is not a way of getting from $G$ to $H$ at all: it's literally a homomorphism from $H$ to $G$! But the axioms of a category are still satisfied in this case, and in many other useful cases where morphisms are not functions.
To help your intuition about what can be a category-that is, anything that satifies the axioms-you might also look into some finite categories, such as $\cdot \to \cdot$. Here the dots have no inherent identity at all-they're just two unspecified objects with an unspecified relationship between them. 
Now, very many categories are naturally concrete, that is, their morphisms are functions, and many more can be made concrete. For instance, my finite category above is isomorphic to the subcategory $\{\emptyset\}\to \{*\}$ of the category of sets. But some categories are provably not equivalent to any concrete category, so this thinking straight from the axioms about morphisms is unavoidable.
