There are a few auxiliary concepts we’ll need before the next major topic. Let’s start with two categories 
and 
, and the category of functors 
. If the following seems very complicated, consider to be any particular toy category you’d like, so this category of functors is a diagram category.
Now, for every object 
there is a constant functor that sends every object of to 
, and every morphism to the identity on . Actually, this assignment of the constant functor to an object of is a functor from to . Indeed, given a morphism 
we get a natural transformation from the one constant functor to the other, whose component at each object of is 
. We call this the “diagonal” functor 
. That is, 
is the constant functor with value .
Let 
be some particular functor in . A cone is an object in the comma category 
. Let’s unpack this definition. Since 
has as its source, and 
is a fixed object — the same thing as a functor from the category 
to , we know what objects of this category look like.
An object of the category consists of an object of and an arrow from to . But an arrow in is a natural transformation of functors. That is, for each object 
of we need an arrow in from ](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%5CDelta%28C%29%5Cright%5D%28J%29&bg=e6e6e6&fg=000000&s=0 "\left[\Delta(C)\right](J)")
to 
. But =C](http://l.wordpress.com/latex.php?latex=%5Cleft%5B%5CDelta%28C%29%5Cright%5D%28J%29%3DC&bg=e6e6e6&fg=000000&s=0 "\left[\Delta(C)\right](J)=C")
for all objects . So we just need an object and an arrow 
for each 
.
Of course, there’s also naturality conditions to be concerned with. If 
is an arrow in , and 
and 
are the required arrows from , then naturality requires that 
. So we need a collection of arrows from to the objects in the image of that are compatible with the arrows from . Such a collection defines an object in the comma category — a cone on .
Cocones are defined similarly. A cocone on is an object in the comma category 
. That is, it’s an object and a collection of arrows from the objects in the image of that are compatible with the arrows from .
The description may seem a little odd, but try writing it out for some very simple categories . For example, let be a set. Then try letting it be an ordinal, or another preorder. After you write down the definition of a cone and a cocone on some simple categories the general idea should seem to make more sense.
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2 ‘formula does not parse’ latex errors here
Yeah, those have been creeping in. WordPress changed something in their parser and now brackets give trouble unless they’re given as \left[ \right] pairs. I don’t know which pages have the problems, but when I stumble across them I fix them. Thanks for pointing this example out to me.