Kan extension of a composition

Question

I'm trying to prove this result, but I'm stuck. Let $f:C\to D$, $g:D\to E$ and $i: C\to B$ be functors and suppose (1) that all the kan extensions i'm about to name exist globally and (2) not sure it is needed but let's say that $i$ is fully faithful. Then $lan_i(gf)\simeq g\circ lan_i(f)$. My strategy so far consists in noticing that we know that $[B,D](lan_i(f), -)\simeq [C, D](f, -\circ i)$ and we want to show that $[B,E](g\circ lan_i(f), -)\simeq [C, E](g\circ f, -\circ i)$ so in some sense this proof would be a one liner of the form: apply the same functor to isomorphic things and you'll get isomorphic things, but I am failing to make this intuition precise

Answer 1

A Kan-extension which has your property is called an absolute Kan extension. Absolute Kan-extension are in particular pointwise Kan-extension, but not all Kan-extensions are pointwise. This means that the property that you are trying to prove will not hold for all Kan-extensions.