A lemma (6.3.8) in Emily Riehl's Category Theory in Context (the Kan Extension chapter) states (the dual of) the proposition
Given functors $F: \mathscr{C} \to \mathscr{E}$ and $K: \mathscr{C} \to \mathscr{D}$, with $\mathscr{D}$ and $\mathscr{E}$ locally small, and an object $d \in \mathscr{D}$, from which we define $\Pi_d: K \downarrow d \to \mathscr{C}$, there is a natural isomorphism $$\operatorname{Cocone}(F \Pi_d,e) \cong \mathsf{Set}^{\mathscr{C}^\text{op}} (\mathscr{D}(K-,d), \mathscr{E}(F-,e))$$
Question: Is there a sense in which this is natural in $d$?
I can see how the right hand side could be acted on by $\mathscr{D}$ morphisms, but on the left hand side, I'm not sure how a $\mathscr{D}$ morphism would get me from $$\operatorname{Cocone}(F \Pi_d,e) \quad\text{to} \quad \operatorname{Cocone}(F \Pi_{d'},e).$$
A map $d \xrightarrow{g} d'$ surely gives a functor $K \downarrow d \to K \downarrow d'$, but the diagram $$\begin{array} && K \downarrow d & \xrightarrow{g_*}& K \downarrow d' \\ &\Pi_d&\searrow &\swarrow&\Pi_d' \\ &&\mathscr{C} \end{array} $$
seems not to commute, (Edit: yes it does) so I can't quite see how to make it work.
She may not have meant naturality in $\mathscr{D}$, but I am interested in whether it is there, nevertheless.
Edit: I think that triangle does commute. I'm going to endeavor to answer my own question here in a while.
On the one hand, the data of a cocone with vertex $e$ in $\mathscr E$ under the diagram $(K\downarrow d)\xrightarrow{\Pi_d}\mathscr C\xrightarrow{F}\mathscr E$ associates to every morphism $Kc\xrightarrow{\phi} d$ a morphism $Fc\xrightarrow{\phi'}e$, so that $Kc_1\xrightarrow{Kf}Kc_2\xrightarrow{\phi}d$ is associated to $F{c_1}\xrightarrow{Ff}Fc_2\xrightarrow{\phi}d$. In other words, so that $(\phi\circ Kf)'=\phi'\circ Ff$.
On the other hand, this is precisely the data of a natural transformation $\mathscr D(K-,d)\Rightarrow\mathscr E(F-,e)$ since the following diagram is asserted to commute for every $c_1\xrightarrow{f}c_2$ in $\mathscr C$. $$\require{AMScd}\begin{CD} \mathscr D(Kc_2,d) @>\phi\mapsto\phi'>>\mathscr E(Fc_2,e)\\ @V\mathscr D(Kf,d)VV @V\mathscr E(Ff,e)VV\\ \mathscr D(Kc_1,d) @>\psi\mapsto\psi'>>\mathscr E(Fc_2,e) \end{CD}$$
Now, consider a morphism $d_1\xrightarrow{g}d_2$ in $\mathcal D$ and a fixed cocone under $(K\downarrow d_2)\xrightarrow{\Pi_{d_2}}\mathscr C\xrightarrow{F}\mathscr E$. On the one hand, the morphism$d_1\xrightarrow{g}d_2$ determines a functor $(K\downarrow d_1)\xrightarrow{g_*}(K\downarrow d_2)$ sending $Kc\to d_1$ to the composite $Kc\to d_1\xrightarrow{g}d_2$; functoriality follows from the fact that pre- and post-composition commute, e.g. $Kc_1\xrightarrow{Kf}Kc_2\to d_1$ gets sent to $Kc_1\xrightarrow{Kf}Kc_2\to d_1\xrightarrow{g}d_2$. We also see that the diagram of functors
$$\begin{CD} (K\downarrow d_1) @>g_*>> (K\downarrow d_2)\\ @V\Pi_{d_1}VV @V\Pi_{d_2}VV\\ \mathscr C @= \mathscr C\\ @VKVV @VKVV\\ \mathscr D @= \mathscr D \end{CD}$$ commutes, giving us an action on the cocones. Explicitly, $Kc\xrightarrow{\phi}d_1$ gets sent to $Fc\xrightarrow{\phi''}e$ defined as where the cocone on $d_1$ sends $Kc\xrightarrow{\phi}d_1\xrightarrow{g}e$. In other words, $\phi''=(g\circ\phi')$, hence we indeed have a cocone by since the equation $(\phi\circ Kf)''=(g\circ\phi\circ Kf)'=(g\circ\phi)'\circ Kf=\phi''\circ Kf$.
So yes, the isomorphism $
On the other hand, this obvious action on cocones is precisely pre-composition with the natural transformation $\mathscr D(K-,d_1)\xrightarrow{\mathscr D(K-,g)}\mathscr D(K-,d_2)$ to produce $\mathscr D(K-,d_1)\xrightarrow{\mathscr D(K-,g)}\mathscr D(K-,d_2)\Rightarrow\mathscr E(F-,e)$. Note that the equation above becomes the commutativity of the diagram $$\require{AMScd}\begin{CD} \mathscr D(Kc_2,d_1) @>\mathscr D(Kc_2,g)>> \mathscr D(Kc_2,d_2) @>\phi\mapsto\phi'>>\mathscr E(Fc_2,e)\\ @V\mathscr D(Kf,d_1)VV @V\mathscr D(Kf,d_2)VV @V\mathscr E(Ff,e)VV\\ \mathscr D(Kc_1,d_1) @>\mathscr D(Kc_1,g)>>\mathscr D(Kc_1,d_2) @>\psi\mapsto\psi'>>\mathscr E(Fc_2,e) \end{CD}$$
So yes, the isomorphism $\operatorname{Cocone}(F \Pi_d,e) \cong \mathsf{Set}^{\mathscr{C}^\text{op}} (\mathscr{D}(K-,d), \mathscr{E}(F-,e))$ is indeed natural.