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Part of my intuition about the construction of limits and colimits is based on the idea that they are initial and terminal objects in the appropriate category: The limit of a diagram $D$ is of course a/the final object in the category of cones over $D$, and similarly for colimits and cocones. The part that I don't understand is that there seem to be four options here, and only two are commonly discussed:

$$\begin{array}{c|cc} & \text{Terminal} & \text{Initial} \\ \hline \text{Cones} & \text{Limit} & ?? \\ \text{Cocones} & ?? &\text{Colimit} \end{array}$$

What fills in the blanks, and why are they less emphasized? It seems like we want final objects when arrows are naturally going 'out of something' and initial objects when they're going 'into something', which vaguely makes sense but doesn't feel satisfactory. Products are of course important. On the other hand, nothing prevents one from asking for an initial object in some category of spans, but books (as far as I've read, which only involves the basics) never mention them. Is there a reason?

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    $\begingroup$ This is basically a special case of this question. Terminal cones aren't useful just because they are terminal, they are useful because they are linked to representability in an essential way. $\endgroup$ – user54748 Jan 4 '15 at 4:40
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    $\begingroup$ In a category with initial (respectively final) objects, they are not interesting, because the initial object is automatically the apex of an initial cone for every diagram. $\endgroup$ – hmakholm left over Monica Jan 4 '15 at 5:07
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    $\begingroup$ On the other hand, for example, in the category of fields, an initial cone over a one-point diagram is the prime subfield, which is not uninteresting. $\endgroup$ – hmakholm left over Monica Jan 4 '15 at 5:13
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    $\begingroup$ @henning That is because the category is not connected but each connected component has an initial object (so this is in some sense a locally initial object). $\endgroup$ – Tobias Kildetoft Jan 4 '15 at 9:07
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    $\begingroup$ @HenningMakholm: If I interpret "cones over a one-point diagram" correctly, then this just sounds like "Slice Category". $\endgroup$ – Thorsten Jan 6 '15 at 6:50
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Here's one of the several equivalent ways of thinking about limits and colimits in which this question doesn't arise: for a diagram $J$ and a category $C$, there is a natural diagonal functor $C \to C^J$. If limits of shape $J$ always exist in $C$, they organize themselves into a right adjoint for this functor, and if colimits of shape $J$ always exist in $C$, they organize themselves into a left adjoint for this functor. And that's it.

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If a category has an initial object $0$, then every diagram has unique cone with tip $0$, and this cone is initial among all cones. Thus, they are not interesting. Terminal cones are the interesting ones.

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