# Problem with Steve Awodey's "Category Theory" 8.5 Limits in categories of diagrams

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!

• 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… Jul 21, 2021 at 12:27
• 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. Jul 21, 2021 at 15:24
• Generally speaking, the book "Handbook of categorical algebra" (Vol. 1) by Borceux is quite detailed. Jul 21, 2021 at 15:28

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.