Are Kan extensions extensions in the traditional sense? Suppose $A$ is a full subcategory of $B$. Given $F:A\to Z$, is it true that the left Kan extension agrees with $F$ on $A$, ie. for all $a\in A$, $\mathrm{Lan}_i F(a)\simeq F(a)$, where $i:A\to B$ is the inclusion?

You might be interested in these notes I wrote a few months ago. When I say "It turns out that this definition, albeit correct, it too general" I mean precisely what is contained in Kevin's answer!
