Demonstrating why category arrows are best thought of as arrows and not functions
Demonstration
Consider the category, [ilmath]\mathscr{C} [/ilmath], whose objects are finite sets and whose arrows, [ilmath]\xymatrix{A \ar[r]^f & B} [/ilmath] are functions^{[1]}:
 [ilmath]f:A\times B\rightarrow\mathbb{R} [/ilmath]^{[1]}^{[Note 1]} with no imposed conditions.B
We define composition of arrows, given [ilmath]\xymatrix{A \ar[r]^f & B \ar[r]^g & C } [/ilmath], as:
 [ilmath]g\circ f:A\times C\rightarrow \mathbb{R} [/ilmath] as [ilmath]g\circ f:(a,c)\mapsto\sum\{f(a,y)g(y,c)\ \vert\ y\in B\}[/ilmath]^{[1]}
In words, [ilmath]g\circ f[/ilmath] maps [ilmath](a,c)[/ilmath] to the sum of (the products of the form [ilmath]f(a,y)g(y,c)[/ilmath] for any [ilmath]y\in B[/ilmath])
 Claim: this is indeed a category^{[1]}.
Proof of claims
There's only one claim really.
The message provided is:
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Discussion
You may think for a moment "I've seen this" and think Markov chains, or something to do with matrix multiplication, in truth it's just an example.
As required for a category, we can compose arrows, but for a change (even though the objects are sets, which isn't a change from what we usually deal with) composition of arrows isn't straight forward composition of functions.
Notes
 ↑ Yes, this is a function, but wait and you'll see this is quite different to the "usual" kind of arrows!
References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} An Introduction to Category Theory  Harold Simmons  1st September 2010 edition
