If $A$ is a cocomplete category and $C$ is small, we can say that for all functor $C \stackrel{l}{\to} A$ there exists the Kan Extension Lan$\_{y\_C}(l): \hat{C} \to A$. Moreover, the Kan extension is pointwise.


Is it known if the other implication is true? Namely:



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> Conj: If for all functor $C \stackrel{l}{\to} A$ there exists the Kan Extension Lan$\_{y\_C}(l): \hat{C} \to A$ then $A$ is cocomplete.
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Is this a characterization of cocompleteness?
Then answer is yes if you assume the extensions are pointwise.


Let $\ast$ be the terminal presheaf on $\hat C$. Because the extension is poinwise, we have $Lan\_{y\_C}(l)(\ast) = \varinjlim\_{c \in C} l(c)$. This can be seen, for example, by calculating the colimit using the category of elements of $\ast$, which is just $C$ itself.


