When is an isomorphism between representable functors natural?

Question

Let $\mathcal{A},\mathcal{B}$ be two small categories and $\mathcal{C},\mathcal{D}$ two arbitrary categories. Let $F:\mathcal{A}\rightarrow\mathcal{B}$, $G:\mathcal{A}\rightarrow\mathcal{C}$ and $L:\mathcal{C}\rightarrow\mathcal{D}$. Suppose the Kan extension of $G$ along $F$ exists. Furthermore suppose that for each $H\in[\mathcal{B},\mathcal{D}]$, we have the following isomorphism: $$\text{Nat}(L\circ\text{Lan}_F(G),H)\cong\text{Nat}(\text{Lan}_F(L\circ G),H).$$ When is this isomorphism natural in $H$? I.e., when do we have a natural isomorphism of the form $$\theta:\text{Nat}(L\circ\text{Lan}_F(G),-)\cong\text{Nat}(\text{Lan}_F(L\circ G),-)?$$ What is the relation between Yoneda and this? For example, I know that by Yoneda we have the following isomorphism $$\text{Nat}(\text{Nat}(L\circ\text{Lan}_F(G),-),\text{Nat}(\text{Lan}_F(L\circ G),-))\cong\text{Nat}(\text{Lan}_F(L\circ G),L\circ\text{Lan}_F(G))$$ but I am not sure what use this has.