- Let's take a functor category $[X, Y]$.
- Take a diagram $D: J \to [X, Y]$. What is the limit $\lim D: [X, Y]$?
- First, let's assume that $X$ has no arrows, or that we forget all the arrows of $X$ except the identity arrows. denote this forgotten/discrete category by $ob(X)$, whose objects are those of $X$, and morphisms are only identity morphisms.
- We can define the diagram $ob(D): J \to [ob(X), Y]$. Can we compute $\lim ob(D)$?
- A functor $ob(X) \to Y$ is the same as a tuple $Y^{ob(X)}$. See that $Y^{ob(X)}$ lives in
`CAT`

, since it is a category that is the $ob(X)$ copies of $Y$.

- Now, the limit $ob(D)$ can be interpreted as a limit of $ob(D): J \to Y \times Y \times \cdots \times Y$.
- By the universal property of the product, limits over product categories can be computed
*pointwise*. So if we have a diagram $E: K \to X \times Y$, then $l \equiv \lim E$ can be calculated by calculating $l_x \equiv \lim (\pi_1 \circ E : K \to X)$, then $l_y \equiv \lim (\pi_2 \circ E : K \to Y)$, and then setting $l \equiv (l_x, l_y) \in X \times Y$. - Thus, we split the morphism $ob(D): J \to Y \times Y \times \cdots \times Y$ into the individial tuple components, which correspond to the images of $x \in ob(X)$ under $D$, and we compute their limits. So we can compute this pointwise.

- Suppose we had
`J = (f -a-> h <-b- g)`

, and we had`ob(X) = (p q)`

. We only have objects, no morphisms. - Now, what is a diagram
`ob(D): J -> [ob(X), Y]`

? For each of`f, g, h`

in`J`

, we must get a functor from`ob(X)`

to`Y`

. - Denote
`F = ob(D)(f)`

,`G = ob(D)(g)`

, and`H = ob(D)(h)`

. Each of`F, G, H`

are functors`ob(X) -> Y`

. - I'll write the functors by identifying them by their image. The image of
`F`

is going to be`[Fp Fq]`

with no interesting morphisms between`Fp`

and`Fq`

. - Now, that we've considered the action of
`ob(D)`

on objects of`J`

, what about the arrows? - The images of the arrows
`f -a-> h`

and`h <-b- g`

are natural transformations from`F`

to`H`

and`G`

to`H`

respectively. Denote these by`F =α>= H`

and`H <=β=G`

. So we have`ob(D)(a) = α`

,`ob(D)(b) = β`

. - In total, the image of
`ob(D)`

in`[ob(X), Y]`

looks like this:

```
F =α=> H <=β= G
```

- If we expand out the functors by identifying them with the image, and write the natural transformations in terms of components, it looks like so:

```
[Fp Fq]
| |
αp αq
v v
[Hp Hq]
^ ^
βp βq
| |
[Gp Gq]
```

- Really, the diagram consists of two parts which don't interact: the part about
`p`

and the part about`q`

. So computing limits should be possible separately!

`[X, Y]`

- We now believe that given $D: J \to [X, Y]$, we know that we can compute $ob(D): J \to [ob(X), Y]$ pointwise.
- Formally, we define $[\lim ob(D)](x)$ to be equal to $\lim (ev_x \circ D : J \to Y)$.
- We define the action of $\lim D$ (which is a functor from $X$ to $Y$) on objects of $X$ to be equal to the action of $\lim ob(D)$ on objects of $X$, which is given by the above equation.
- So what about the action of $\lim D$ on the
*morphisms*of $X$? it's a functor from $X$ to $Y$, so it should send morphisms to morphisms! - Now, let's suppose we have a morphism $x \xrightarrow{a} x'$ in $X$. How do we compute the the action of $D$ on the morphism $a$?
- Well, first off, what's $D(a)$ a morphism between? It must be between $D(x)$ and $D(x')$.
- What is $D(x)$? We know that $D(x) \equiv \lim (ev_x \circ D: J \to Y)$. Similarly, we know that $D(x') \equiv \lim ev_x' \circ D: J \to Y)$.