I trying to decide whether the existence of a, say, a left Kan extension along $i: C \to D$ for any functor $F: C \to E$ implies that $i$ has a right adjoint.
I have proven the converse, and I know that if left Kan extensions existed and were absolute, i.e. preserved by any functor $G: E \to E'$, then $Ran_{F}id_C$ would be the desired adjoint.
I initially looked for a counterexample, but this turned out to be unsuccessful. I really have no idea how to approach this unfortunately, so any help is appreciated!