Limit of $h_{\mathcal C} \circ F$ for $F: I \to \mathcal C$ exists in $\text{Func}(\mathcal C^\text{op}, \text{Set)}$ and is $\text{Hom}(\_,\lim F)$.

Question

Let $I$ be a small category and $\mathcal C$ a complete category. For $F: I \to \mathcal C$, show that the limit of the functor $h_{\mathcal C} \circ F$ exists in $\operatorname{Func}(\mathcal C^{\text{op}}, \text{Set})$ and is represented by $\lim F$.

So I managed to show that $\operatorname{Hom}(\_, \lim F)$ is a cone. However, I am having trouble showing that it is a limiting cone, specifically finding a unique map from any other $G: \mathcal C^{\text{op}} \to \text{Set}$ (with any pairs of isomorphism $ \delta_i, \delta_j$ going from $G$ to $\operatorname{Hom}(\_,F(i))$ and $\operatorname{Hom}(\_,F(j))$ that makes things commute) to $\operatorname{Hom}(\_,\lim F)$. (I won't draw the diagram because it gets messy and because it's obvious what I am referring to).

It seems like I have to do some sort of argument referring back to the diagram corresponding to the $\lim F$, but I can't seem to figure it out. Any help would be great.

Answer 1

$\newcommand\cat{\mathscr}\DeclareMathOperator\op{op}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Set{Set}\DeclareMathOperator\Func{Func}$Let $L=\lim F$ and let $\lambda:L|\cat I\to F$ be a limit cone over $F$.

Let $G:\cat C^{\op}\to\Set$ be a functor and $\tau:G|\cat I\to h\circ F$ be a natural transformation between functors $\cat I\to\Func(\cat C^{\op},\Set)$.

For all $I\in\cat I$ we have a natural transformation $\tau_I:G\to\Hom_{\cat C}(-,F(I))$, hence for all $X\in\cat C$ we have a function $(\tau_I)_X:G(X)\to\Hom_{\cat C}(X,F(I))$. Pick an element $a\in G(X)$, the family of morphisms $(\tau_I)_X(a):X\to F(I)$ is natural respect to $I$, hence there exists one and only one morphism $\sigma_X(a):X\to L$ such that $(\tau_I)_X(a)=\sigma_X(a)\lambda_I$ for all $a\in G(X)$.

This defines a function $\sigma_X:G(X)\to\Hom_{\cat C}(X,L):a\mapsto\sigma_X(a)$ which is natural in $X$. For if $f:X\to Y$ is a morphism in $\cat C$ then $\require{AMScd}$ \begin{CD} G(Y)@>\sigma_Y>>\Hom_{\cat C}(Y,L)\\ @VG(f)VV @VV\Hom_{\cat C}(f,L)V\\ G(X)@>>\sigma_X>\Hom_{\cat C}(X,L)\\ \end{CD}

Thus $\sigma:G\to h(L)$ is a natural transformation between functors $\cat C^{\op}\to\Set$ and $\tau_I=\sigma h(\lambda_I)$ for all $I\in\cat I$.