First, I state premilinary results.
For a presheaf $X\colon A^{op}\to\mathsf{Set}$, it's category of elements, denoted by $\int X$, has pairs $(a,s)$ where $a \in A$ and $s \in X(a)$ as objects and $f\colon a\to b$ such that $X(f)(t) = s$ as morphisms $(a,s)\to (b,t)$.
Cisinski, Proposition 1.1.8 For each presheaf $X$ over $A$, let the functor $\phi_X\colon \int X\to [A^{op},\mathsf{Set}]$ be the composition of the forgetful functor $\prod_X\colon \int X\to A$ and the Yoneda embedding $Y_A\colon A\to [A^{op},\mathsf{Set}]$. Define a cocone $\lambda^X\colon \phi_X\Rightarrow X$ given by $(\lambda^X)_{(a,s)} = y^{-1}_{a,X}(s)$ where $y_{a,X}\colon\mathsf{Hom}_{[A^{op},\mathsf{Set}]}(\mathsf{Hom}_A(-,a),A)\to A_a$ is the natural bijection from the Yonede lemma. Then $\lambda^X$ is a colimit cocone.
Cisinski, Proposition 1.1.10 Let $A$ be a small category, $C$ a cocomplete locally small category and $u\colon A\to C$ a functor. For each presheaf $X$ over $A$, define a functor $u_X\colon \int X\to C$ given by $u_X(a,s) = u(a)$ and $u_X(f) = u(f)$. For each presheaf $X$, choose a colimit $L_X$ and a colimit cocone $\mu^X\colon u_X\Rightarrow L_X$. Define a functor $u_!\colon [A^{op},\mathsf{Set}]\to C$ by making it send a presheaf $X$ to $L_X$ and a morphism $f\colon X\Rightarrow Y$ of presheaves to the unique morphism $u_!(f)\colon L_X\to L_Y$ such that $u_!(f)\circ\mu^X_{(a,s)} = \mu^Y_{(a,u_a(s))}$ for any $(a,s) \in \int X$. Then $u_!$ has a right adjoint (for detalis about this proposition, see this question).
Finally, the remark in question.
Cisinksi, Remark 1.1.11 The functor $u_!$ will be called the extension of $u$ by colimits. In fact, any cocontinuous functor $F\colon [A^{op},\mathsf{Set}]\to C$ is isomorphic to the functor of the form $u_!$. More precisely, if we put $u(a) = F(\mathsf{Hom}_A(-,a))$ and $u(f) = F(\mathsf{Hom}_A(-,f))$, there is a unique natural isomorphism $u_!(X) \cong F(X)$ which is the identity whenever the presheaf $X$ is representable.
I've constructed a natural isomorphism $\eta\colon u_!\Rightarrow F$ by setting $\eta_X\colon u_!(X)\to F(X)$ be the unique morphism for which we have $\eta_X\circ \mu^X_{(a,s)} = F(\lambda^X_{(a,s)})$ for any $(a,s) \in \int X$ ($\mu^X$ and $\lambda^X$ mean what they meant above).
My question is:
Is my natural isomorphism gives identity when $X$ is representable? If so, why?
If not, what is the correct natural isomorphism?
At any case, how to prove uniqueness of said natural isomorphism which gives identity whenever whenever it's value is a representable presheaf?