Consider functors $F:C\to D$ and $G:C\to C'$. It is straightforward to prove that the left Kan extension of $F$ along $G$, denoted by $Lan_G F$, is functorial in $F$.
How about in the argument $G$, instead? I'm looking for a statement in the following form.
Let $\phi:G\to G'$ be a natural transformation between functors $G,G':C\to C'$. We can obtain canonically a natural transformation $Lan_{G'}F\to Lan_G F$ between functors $C'\to D$.
Note the direction of the arrow, I expect a contravariant functor.
If this holds, I'm sure it has been done somewhere. Where can I find a reference?