So if $\mathcal{C}$ is locally small, we have the Yoneda embedding $Y:\mathcal{C} \rightarrow [\mathcal{C}^{op},Sets ]$. This preserves all limits in $\mathcal{C}$, and a comment here: An application of Yoneda Lemma says that showing this is just a matter of reformulating the definition of a limit. How is this?
I thought about how I would show that $Y$ preserves limits, and all I could think of was this: Suppose $L$ is the limit in $\mathcal{C}$ of some diagram $D$. Take some cone over $YD$ in $[\mathcal{C}^{op},Sets]$ with base $P$ and write $P$ as a colimit of representable presheaves. (I think) you can then use the Yoneda embedding to show that $L$ is a cocone over the elements of $P$, and the unique map $P \rightarrow YL$ from the universal property of $P$ will also work to show that $YL$ is indeed the limit.
Is this proof correct, and what is the easy proof that I've missed?
Best Answer
If $\{L \to D_i\}$ is a limit diagram in $C$, then for every $X \in C$ we have that $\{\hom(X,L) \to \hom(X,D_i)\}_i$ is a limit diagram in $\mathsf{Set}$, which means precisely that $\{\hom(-,L) \to \hom(-,D_i)\}_i$ is a limit diagram in $[C^{op},\mathsf{Set}]$.