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Proposition 8.7. For any locally small category $\mathbb{C}$, the functor category $\mathbb{Sets}^{\mathbb{C}^{op}}$ is complete. Moreover, for every object $C \in \mathbb{C}$, the evaluation functor $$ev_C:\mathbb{Sets}^{\mathbb{C}^{op}} \to \mathbb{Sets}$$ preserves all limits.

Proof. Suppose we have $J$ small and $F:J \to \mathbb{Sets}^{\mathbb{C}^{op}}$. The limit of $F$, if it exists, is an object in $\mathbb{Sets}^{\mathbb{C}^{op}}$, hence is a functor, $$(\lim_{j \in J}F_j):\mathbb{C}^{op} \to \mathbb{Sets}$$ By the Yoneda lemma, if we had such a functor, then for each object $C \in \mathbb{C}$ we would have a natural isomorphism, $$(\lim F_j)(C) \cong \mathrm{Hom}(yC,F_j)$$ But then it would be the case that $$ \mathrm{Hom}(yC, \lim F_j) \cong \lim \mathrm{Hom}(yC, F_j) \cong \lim F_j(C) $$ in $\mathbb{Sets}$ where the first isomorphism is because representable functors preserve limits, and the second is Yoneda again. Thus, we are led to define the limit $\lim_{j \in J}F_j$ to be $$(\lim_{j \in J}F_j)(C)=\lim_{j \in J}(F_jC) \tag{8.4}$$ that is, the pointwise limit of the functors $F_j$. The reader can easily work out how $\lim F_j$ acts on $\mathbb{C}-arrows$, and what the universal cone is, and our hypothetical argument then shows that it is indeed a limit in $\mathbb{Sets}^{\mathbb{C}^{op}}$.

Finally, the preservation of limits by evaluation functors is stated by (8.4).

I'm having some trouble typing the limit signs, so I ignored the $\leftarrow$s below the $\lim$s because I don't know how to stack two lines of subscripts below the limit signs.

I'm confused with 'our hypothetical argument then shows that it is indeed a limit'. Why does this follow from the hypothetical argument? Why isn't there a general cone in $\mathbb{Sets}^{\mathbb{C}^{op}}$ of which the universal mapping property of the limit should be verified?

Thanks!

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  • $\begingroup$ The first half of the sentence explicitly says that the reader can work out what the functor and cone looks like. The hypothetical argument is not a proof, but intended to be a hint why we might expect that the limit exists and how it may be computed… $\endgroup$ Jul 21, 2021 at 12:27
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    $\begingroup$ I suggest to read Mac Lane or other category theory books on this lemma. What Awodey presents here as a proof is just it's motivation, I don't see anything which can count as a proof. Also, the sentence "and our hypothetical argument then shows that it is indeed a limit" is not correct unless you add a fact about density, namely that every functor is a canonical colimit of representables, which explains why we only need to restrict to these functors when testing the universal property of the limit. $\endgroup$ Jul 21, 2021 at 15:24
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    $\begingroup$ Generally speaking, the book "Handbook of categorical algebra" (Vol. 1) by Borceux is quite detailed. $\endgroup$ Jul 21, 2021 at 15:28

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This is certainly missing from the proof, but here's a justification.

Observation: due to Yoneda, it suffices to check limits in a functor category on the representables.

Why? Suppose $F\to F_j$ is a cone which satisfies the universal property for maps out of representables $\newcommand\C{\mathcal{C}}\C(-,A)$. Suppose $X\to F_j$ is a cone. Let $a\in X(A)$. By Yoneda, this gives a map $\C(-,A)\to X$, and thus a cone $\C(-,A)\to X\to F_j$. By assumption, there is a unique map $\C(-,A)\to F$ compatible with this cone, corresponding to some element $\tau_A(a)\in F(A)$. Define a map $\tau_A : X(A)\to F(A)$ by $a\mapsto \tau_A(a)$.

Then you can check that the $\tau_A$ are natural as $A$ varies, since if $f : A\to B$, $\tau_B(a|_f)$ corresponds to the map $\C(-,B)\xrightarrow{f^*} \C(-,A)\to X\to F$, which is also what $f^*\tau_A(a)$ corresponds to. Thus we have a map of functors $\tau : X\to F$.

It's clear by construction that $X\xrightarrow{\tau} F\to F_j$ gives our original cone back, and any map $X\to F$ compatible with the cones must send $a\in X(A)$ to $\tau_A(a)$ by the uniqueness property of maps into $F$ on representables.

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