Is it described somewhere in the literature how exponentials in functor categories other than presheaf categories can be constructed? So far I did not find anything in this regard. In the following, I sketch what I have developed myself. I would like to know whether this can be improved, and whether it is similar to some standard approach.

$\newcommand{\Obj}{\mathop{\mathrm{Obj}}}\newcommand{\Mor}{\mathop{\mathrm{Mor}}}\newcommand{\dom}{\mathop{\mathrm{dom}}}\newcommand{\codom}{\mathop{\mathrm{codom}}}\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\hom}{\mathrm{hom}}\newcommand{\id}{\mathrm{id}}\newcommand{\Tw}{\mathrm{Tw}}\newcommand{\op}[1]{#1^{\mathrm{op}}}\newcommand{\Set}{\mathbf{Set}}$In a presheaf category $\mathbf{Set}^{\op{\mathcal C}}$, exponentials are obtained by defining$$G^F(f) = \Hom(Yf \times \id_F, G)\enspace,$$ where $Y$ is the Yoneda embedding. If we turn from $\Set^{\op{\mathcal C}}$ to $\Set^{\mathcal C}$, this definition becomes$$G^F(f) = \Hom(\Hom(f, {-}) \times \id_F, G)\enspace.$$So applying an exponential $G^F$ to an object $C$ yields the set of natural transformations from $\Hom(C, {-}) \times F$ to $G$.

This way of constructing exponentials only works if the target category of the functor category is $\Set$, for the following reasons:

The $\Hom$ functor used in $\Hom(C, {-})$ has the target category $\Set$.

The set of natural transformations from some functor to another one is, well, a set.

Let us now try to generalize the above construction such that it works for functor categories $\mathcal D^{\mathcal C}$ where $\mathcal D$ is not necessarily $\Set$.

We first tackle the second of the above points. Let $F$ and $G$ be functors with $F, G : \mathcal C \to \mathcal D$. We define the functor $M : \Tw(\mathcal C) \to \mathcal D$ from the twisted arrow category of $\mathcal C$ to the category $\mathcal D$ as follows:

For every object $f$ in $\Tw(\mathcal C)$, which is a morphism $f : A \to B$ in $\mathcal C$, we define$$Mf = (GB)^{FA}\enspace.$$

For every morphism $(p, q) : f \to g$ in $\Tw(\mathcal C)$, we define$$M(p, q) = (Gq)^{Fp}\enspace.$$

If the limit of $M$ exists, then this limit represents the set of natural transformations from $F$ to $G$. This can be made plausible by considering the special case that $\mathcal D = \Set$. In this case, the limit of $M$ is the set of all indexed families $\{m_h\}_{h \in \Mor \mathcal C}$ that have the following properties:

If $h$ is a morphism in $\mathcal C$ with $h : A \to B$, then $m_h$ is a function from $FA$ to $GB$.

If $f$, $g$, and $h$ are morphisms in $\mathcal C$ with $f : A' \to A$, $g : B \to B'$, and $h : A \to B$, then$$m_{ghf} = (Gg)m_h(Ff)\enspace.$$

Each indexed family $\{m_h\}$ gives rise to a natural transformation $\tau : F \to G$ with $\tau_A = m_{\id_A}$ for every $A \in \Obj \mathcal C$. To see this, let us look again at the above two properties of families $\{m_h\}$:

The first property tells us that for each object $A$, we have $m_{\id_A} : FA \to GA$. This corresponds to $\tau_A : FA \to GA$.

From the second property, we can deduce that the following equations must hold for any morphism $h : A \to B$:$$\begin{align}m_{h\id_A\id_A} & = (Gh)m_{\id_A}\id_{FA}\\m_{\id_B\id_Bh} & = \id_{GB}m_{\id_B}(Fh)\end{align}$$This in turn gives us$$(Gh)m_{\id_A} = m_{\id_B}(Fh)\enspace,$$ which corresponds to the naturality condition$$(Gh)\tau_A = \tau_B(Fh)\enspace.$$

It is relatively easy to show that the mapping from families $\{m_h\}$ to natural transformations $\tau : F \to G$ is actually a bijection.

Now we do not want to represent the set of natural transformations from $F$ to $G$, but from $\Hom(C, {-}) \times F$ to $G$ for any $C \in \Obj \mathcal C$. To do so, we replace the functor $M : \Tw(\mathcal C) \to \mathcal D$ by a functor $L : \Tw(C \downarrow \mathcal C) \to \mathcal D$, which we define as follows:

For any morphisms $a : C \to A$ and $b : C \to B$ in $\mathcal C$, which are objects in $(C \downarrow \mathcal C)$, and any morphism $f : A \to B$ in $\mathcal C$ with $fa = b$, which is a morphism from $a$ to $b$ in $(C \downarrow \mathcal C)$ and thus an object in $\Tw(C \downarrow \mathcal C)$, we define$$Lf= (GB)^{FA}\enspace.$$

For any morphism $(p, q) : f \to g$ in $\Tw(C \downarrow \mathcal C)$, we define$$L(p, q) = (Gq)^{Fp}\enspace.$$

If the limit of $L$ exists, then it represents the set of natural transformations from $\Hom(C, {-}) \times F$ to $G$. To make this plausible, we consider again the case that $\mathcal D = \Set$. In this case, the limit of $L$ is the set of all indexed families $\{\ell_{a, h}\}_{a, h \in \Mor \mathcal C \wedge \codom a = \dom h}$ that have the following properties:

If $a$ and $h$ are morphisms in $\mathcal C$ with $a : C \to A$ and $h : A \to B$, then $\ell_{a, h}$ is a function from $FA$ to $GB$.

If $a'$, $f$, $g$, and $h$ are morphisms in $\mathcal C$ with $a' : C \to A'$ $f : A' \to A$, $g : B \to B'$, and $h : A \to B$, then$$\ell_{a', ghf} = (Gg)\ell_{fa', h}(Ff)\enspace.$$

Each indexed family $\{\ell_{a, h}\}$ gives rise to a natural transformation $\tau : \Hom(C, {-}) \times F \to G$ with $\tau_A(a, x) = \ell_{a, \id_A}(x)$ for any $A \in \Obj \mathcal C$, $a : C \to A$, and $x \in FA$. Let us look at the above two properties of families $\{\ell_{a, h}\}$ to see why this is the case:

The first property tells us that for each object $A$ and each morphism $a \in \hom(C, A)$, we have $\ell_{a, \id_A} : FA \to GA$. This corresponds to $\tau_A : \Hom(C, A) \times FA \to GA$.

From the second property, we can deduce that the following equations must hold for any morphisms $a : C \to A$ and $h : A \to B$:$$\begin{align}\ell_{a, h\id_A\id_A} & = (Gh)\ell_{a, \id_A}\id_{FA}\\\ell_{a, \id_B\id_Bh} & = \id_{GB}\ell_{ha, \id_B}(Fh)\end{align}$$This gives us$$(Gh)\ell_{a, \id_A} = \ell_{ha, \id_B}(Fh)$$and thus$$(Gh)(\ell_{a, \id_A}(x)) = \ell_{ha, \id_B}((Fh)(x))$$for each $x \in FA$. This corresponds to$$(Gh)(\tau_A(a, x)) = \tau_B(ha, (Fh)(x))\enspace.$$Since$$\begin{align}\tau_B(ha, (Fh)(x)) & = \tau_B(\Hom(C, h)(a), (Fh)(x)) \\ & = \tau_B((\Hom(C, h) \times Fh)(a, x)) \\ & = \tau_B(((\Hom(C, {-}) \times F)h)(a, x))\enspace,\end{align}$$we get $$(Gh)\tau_A = \tau_B((\Hom(C, {-}) \times F)h)\enspace,$$which is the naturality condition for $\tau$.

I suppose that the mapping from families $\ell_{a, h}$ to natural transformations $\tau : \Hom(C, {-}) \times F \to G$ is actually a bijection, but I have not checked this yet.