Realization of a limit as a set of natural transformations

Question

Consider the slice category associated to the Yoneda embedding $\mathcal{C}$ $\to$ $\mathcal{PSh(C)}$. Now the lemma says, every presheaf is a colimit of representables. So we resort to Yoneda lemma here. We calculate Hom$_{\mathcal{PSh(C)}}$(colim$_{X \in \mathcal{C_{/F}}}$Hom$_{\mathcal{C}}(\_ ,X), \mathcal{G})$ $\cong$ lim$_{X \in \mathcal{C_{/F}}}$Hom$_{\mathcal{PSh(C)}}$(Hom$_{\mathcal{C}}(\_ ,X), \mathcal{G})$ $\cong$ lim$_{X \in \mathcal{C_{/F}}}\mathcal{G}$(X). Now how to realize lim$_{X \in \mathcal{C_{/F}}}\mathcal{G}$(X) as the set of natural transformations from $\mathcal{F}$ to $\mathcal{G}$ ?

Answer 1

This is what you proved in the question.

You started with a presheaf $\mathcal{F}$, wrote it as a colimit $colim_{X \in \text{yoneda}/\small{\mathcal{F}}} Hom(\_, X)$ of representables, considered the set of natural transformations $Nat(\mathcal{F}, \mathcal{G})$ and showed that this is the same as the $lim_{X \in \text{yoneda}/\small{\mathcal{F}}} \mathcal{G}(X)$, which shows that the limit in question is the set of natural transformations from $\mathcal{F}$ to $\mathcal{G}$.

To realize the limit as $Nat(\mathcal{F}, \mathcal{G})$ (that is to obtain an explicit bijection), chase the isomorphims! Here are the details:

The yoneda lemma says that the category $y/\mathcal{F}$ is isomorphic to the coslice category $1/\mathcal{F}$ where $1$ is the singleton set. Thus,$\text{ }lim_{X \in y/\small{\mathcal{F}}} \mathcal{G}(X) \cong \lim [1/\mathcal{F} \xrightarrow{\text{proj}} \mathcal{C}^{op} \xrightarrow{\mathcal{G}} Set]$

$\cong \{(a_{(X,x)}) \in \prod_{(X, x) \in 1/\mathcal{F}} \mathcal{G}(X) : \forall u : (X, x) \rightarrow (Y, y) \text{ in } 1/\mathcal{F},\text{ }G(u) : a_{(X, x)} \mapsto a_{(Y, y)}\}$

$= \{ (a_{(X,x)}) \in \prod_{(X, x) \in 1/\mathcal{F}} \mathcal{G}(X) : \forall u : X \rightarrow Y \text{ in }\mathcal{C}^{op} \text{ with } \mathcal{F}(u)(x) = y,\text{ }\mathcal{G}(u) : a_{(X, x)} \mapsto a_{(Y,y)}\}$

$= \{(a_{(X,x)}) \in \prod_{(X, x) \in 1/\mathcal{F}} \mathcal{G}(X) : \forall u : X \rightarrow Y \text{ in }\mathcal{C}^{op},\text{ }\mathcal{G}(u) : a_{(X, x)} \mapsto a_{(Y, \mathcal{F}(u)(x))} \text{ }\forall x \in \mathcal{F}(X)\}$

$= \{(a_{(X, x)}) \in \prod_{X \in \mathcal{C}^{op}}\prod_{x \in \mathcal{F}(X)} \mathcal{G}(X) : \forall u : X \rightarrow Y \text{ in }\mathcal{C}^{op},\text{ } \mathcal{G}(u) : a_{(X, x)} \mapsto a_{(Y, \mathcal{F}(u)(x))}\text{ }\forall x \in \mathcal{F}(X)\}$

$= \{ (a_{(X,x)}) \in \prod_{X \in \mathcal{C}^{op}} \mathcal{G}(X)^{\mathcal{F}(X)} : \forall u : X \rightarrow Y \text{ in }\mathcal{C}^{op}, \text{ }\mathcal{G}(u) : a_{(X, x)} \mapsto a_{(Y, \mathcal{F}(u)(x))}\text{ }\forall x \in \mathcal{F}(X)\}$

$\cong \{ (a_X) \in \prod_{X \in \mathcal{C}^{op}} Hom_{Set}(\mathcal{F}(X), \mathcal{G}(X))) : \forall u : X \rightarrow Y\text{ in }\mathcal{C}^{op}, \text{ }\mathcal{G}(u) \circ a_X = a_Y \circ \mathcal{F}(u) \}$

$= Nat(\mathcal{F}, \mathcal{G})$.

Here the composite bijection sends a tuple $(a_{(X, x)})_{(X, x) \in 1/\mathcal{F}}$ to the natural transformation $a = (\mathcal{F}(X) \xrightarrow{a_X} \mathcal{G}(X))_{X \in \mathcal{C}^{op}}$ where $a_X(x) = a_{(X, x)}$.

Edit (24/02/2023, 12:05 am):

I believe that you are actually trying to prove the fact that every presheaf is a colimit of representables (which is possibly why you have mentioned using the Yoneda Lemma at the beginning of the question).

Anyways, we have shown that $\lim_{X \in \text{yoneda}/{\mathcal{F}}} \mathcal{G}(X) \cong Hom_{PSh(\mathcal{C})}(\mathcal{F}, \mathcal{G})$ naturally in the presheaf $\mathcal{G} \in PSh(\mathcal{C})$. Further, you have shown at the beginning that $Hom(colim_{X \in \text{yoneda}/\mathcal{F}} Hom(\_, X), \mathcal{G}) \cong \lim_{X \in \text{yoneda}/{\mathcal{F}}} \mathcal{G}(X)$ naturally in $\mathcal{G} \in PSh(\mathcal{C})$.

Hence, $Hom_{PSh(\mathcal{C})}(colim_{X \in \text{yoneda}/\mathcal{F}} Hom(\_, X), \mathcal{G}) \cong Hom_{PSh(\mathcal{C})}(\mathcal{F}, \mathcal{G})$ naturally in $\mathcal{G} \in PSh(\mathcal{C})$. Therefore, $colim_{X \in \text{yoneda}/\mathcal{F}} Hom(\_, X) \cong \mathcal{F}$ by the Yoneda Lemma.