Let $\mathcal{C}$ be a (sufficiently complete and cocomplete) closed monoidal category with internal hom $[-,-]$. Let $F : \mathcal{A} \to \mathcal{C}$ be a functor obtained as the left Kan extension of $F' : \mathcal{B} \to \mathcal{C}$ along a fully faithful functor $i : \mathcal{B} \to \mathcal{A}$. If $G : \mathcal{A} \to \mathcal{C}$ is another functor I can construct the composite
$$
[F, G] : \mathcal{A}^\mathrm{op} \times \mathcal{A} \to \mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{C}.
$$
Since the value of $F$ on $\mathcal{A}$ depends only on the value of $F'$ on $\mathcal{B}$, it is reasonable to assume that the value of $[F, G]$ on $\mathcal{A}^\mathrm{op} \times \mathcal{A}$ depends only on the value of $[F', G]$ on $\mathcal{B}^\mathrm{op} \times \mathcal{A}$. In fact, thanks to the formula for left Kan extensions,
$$
[F(a\_1), G(a\_2)] \simeq \left[ \operatorname\*{colim}\_{(b \to a\_1) \in i/a\_1} F'(b), G(a\_2) \right] \simeq \operatorname\*{lim}\_{(b \to a\_1) \in i/a\_1} [F'(b), G(a\_2)]. 
$$
This exhibits $[F, G] : \mathcal{A}^\mathrm{op} \times \mathcal{A} \to \mathcal{C}$ as a right Kan extension of $[F', G] : \mathcal{B}^\mathrm{op} \times \mathcal{A} \to \mathcal{C}$ along $\mathcal{B}^\mathrm{op} \times \mathcal{A} \to \mathcal{A}^\mathrm{op} \times \mathcal{A}$.


Is it possible to take this a step further and compute $[F, G]$ in terms of $[F', G \circ i]$? That is, how closely can I mimic the formula $\mathrm{Nat}(F, G) \simeq \mathrm{Nat}(F', G \circ i)$ (where $\mathrm{Nat}$ denotes natural transformations of functors) in my category $\mathcal{C}$ by using the internal hom instead of the hom of functors?



Relationship between Kan extensions and internal hom
The analogous adjunction for the enriched category of functors can be deduced from the adjunction for the ordinary category of functors using the Yoneda lemma.
Indeed, to establish a natural isomorphism
$$[F,G]≅[F',G∘i]$$
it suffices (by the Yoneda lemma) to establish an isomorphism
$$\def\cC{{\cal C}}\cC(V,[F,G])≅\cC(V,[F',G∘i]),$$
natural in $V∈\cC$.
Commuting the functor $\cC(V,-)$ with the limit that computes $[-,-]$, we transform the above relation to
$$\def\Nat{\mathop{\sf Nat}}\Nat(F,G^V)≅\Nat(F',(G∘i)^V).$$
Since the powering $(-)^V$ is computed objectwise, we have $(G∘i)^V=G^V∘i$, so the above isomorphism follows from the ordinary adjunction for Kan extensions applied to $F'$ and $G^V$:
$$\Nat(F,G^V)≅\Nat(F',G^V∘i).$$


