Oh. These formulas are the definition. The pointwise Kan extension is defined to be the (co)limit or (co)end as given there.

Okay, so we neeed to explain this better on the page.

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Why would the method of computation matter?
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<p>It's not meant to say that the computations yield different results, if both exist.</p>
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Also, the definition of a pointwise kan extension isn't even given explicitly, it's only mentioned in a few remarks.
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<p>Please, feel encouraged to help improve the entry. I know (and probably we all know) that it is far from being perfect or even satisfactory. If you feel you care about it, try to improve it., And be it only by adding a query box somewhere saying for instance "At this point such and such should be stated".</p>
<p>Apart from that general remark, isn't the pointwise definition explicitly given in the sections</p>
<p><a href="http://ncatlab.org/nlab/show/Kan+extension#byColimits">pointwise in
terms of colimits</a></p>
<p>and</p>
<p><a href="http://ncatlab.org/nlab/show/Kan+extension#byCoends">pointwise in terms of coends</a></p>
<p>?</p>
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Also, the definition of a pointwise kan extension isn't even given explicitly, it's only mentioned in a few remarks. ]]>