Category with Products and Equalizers also has Limits (Schapira)

Question

In Schapira's text Algebra and Topology, he gives a functorial definition of limits, and states the following theorem (on page 34): If $\alpha:I\rightarrow C$ is a functor ($I$ small and $C$ locally small), and $C$ has both products and equalizers, then the limit of $\alpha$ exists. In fact, if we define $a$ and $b$ such that $$a,b :\prod_i \alpha i \longrightarrow \prod_{s : i \to j}\alpha j$$

where $a$ and $b$ are the unique morphisms such that $$ \pi_{(s : i \to j)}\circ a =\pi_j\\ \pi_{(s : i \to j)}\circ b = (\alpha s)\circ\pi_i $$ then the equalizer object $L$ of $a,b$ is the limit of $\alpha$.

Now Schapira states that it suffices to prove the result with respect to $C=\mathbf{Set}$. Why is this the case?

Answer 1

The answer is that we can use the Yoneda lemma to translate from an arbitrary category $C$ to $\newcommand\Set{\mathbf{Set}}$ through the following pathway.

Fact 1. $\Set$ is complete, and we can compute limits using products and equalizers as claimed.

Now we bootstrap this up to any presheaf category with the following fact:

Fact 2. For any category $C$, the presheaf category $[C,\Set]$ is complete, and limits are computed pointwise. In other words, if $D:J\to [C,\Set]$ is a diagram, then we can produce $\lim D$ by defining $$(\lim D)(c) := \lim_{i\in J} (D(i)(c)).$$

In particular, all small products exist and equalizers exist, and an arbitrary (small) limit can be computed in terms of them as claimed, since everything is computed pointwise.

Now we apply the Yoneda embedding to embed $C$ into $[C,\Set]$, and use the following fact to conclude that for an arbitrary category with all small products and equalizers, all small limits exist and they can be computed as claimed.

Fact 3. If $F:J\to C$ is a small diagram, then a cone $(c,\alpha_i)$ to $F$ determines a morphism $$y_c\to \lim_{i\in J} y_{Fi},$$ whose components are $y_{\alpha_i}$, where $y$ is the Yoneda embedding. Moreover, this morphism is an isomorphism if and only if $(c,\alpha_i)$ is a limit cone. Conversely, such an isomorphism between $y_c$ and $$\lim_{i\in J} y_{Fi}$$ induces a limit cone structure on $c$. (Another way of saying this is that the Yoneda embedding preserves and reflects (small) limits)

What this means is that we can take the object $c$ which is the equalizer of the maps $a$ and $b$ between the products, then $y_c$ is still the equalizer of the maps $y_a$ and $y_b$, but these are the corresponding $a$ and $b$ maps between the products in $[C,\Set]$, so $y_c$ is the limit of the diagram in $[C,\Set]$, by fact 2, so by fact 3, $c$ is in fact the limit of the diagram.

If you have questions about proving the facts themselves, that should probably be a separate question (or is likely answered already in another question), but if you have questions about applying the facts to prove the claim, you can put those in the comments, and I'll try to edit to be more clear.