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:

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.