Limits via universal arrows and functor categories

Question

I would like to understand is some detail the connection between the 2 snippets taken from McLane's book CWM. Namely, I do not follow the connection between the functor $S$ and categories and arrows $C,r,d,u,f'$ and $D$ in the first snippet and the data $t,C,r,\Delta ,J,\tau.\nu$ and $\text{Lim}$ in the second one. I have a complete mess in it.

Answer 1

It is important to note that what is described in the first paragraph is not quite what is described in the second one:

What is described in the first paragraph is an initial object in the comma category $(c \downarrow S)$ where $S: D \rightarrow C$ and $c \in C$. As described, the elements in this category are pairs $(r,u)$ where $r \in D$ and $u: c \rightarrow Sr$. A morphism $(r,u) \rightarrow (d,v)$ is a map $f: r \rightarrow d$ s.t. $v = Sf \circ u$. Drawing a commutative triangle makes it more aparent which this definition makes sense.

In this sense, a universal arrow (as described in CWM $(c,u)$) is an initial object in the category $(c \downarrow S)$ i.e. for any other object $(e,w) \in (c \downarrow S)$ there exists a unique morphism $g$ (as described before s.t. $g: (c,u) \rightarrow (e,w)$ which means that there exists some $g: c \rightarrow e$ making the aforementioned triangular diagram commute.

Dual to the category $(c \downarrow S)$ is the category $(S \downarrow c)$ with the same data as before. In this category, the objects are again pairs $(r,u)$ but here $u: Sr \rightarrow c$. A morphism $(r,u) \rightarrow (d,v)$ is a map $f: r \rightarrow d$ s.t. $u = v \circ Sf$.

What is now done in the second sniplet is describing the universal arrow $(r, \nu)$ as the terminal object of the category $(S \downarrow c)$.

In short, the data in the first sniplet corresponds to the data in the second as follows:

• the category $D$ corresponds to the category $C$
• the category $C$ corresponds to the functor category $\text{Funct}(J,C)$
• the functor $S$ corresponds to the functor $\Delta: C \rightarrow \text{Funct}(J,C)$
• the pair $(r,u)$ corresponds to a pair $(r, \nu)$ where $\nu: \Delta r \rightarrow F$ is a natural transformation i.e. a morphism in $\text{Funct}(J,C)$
• the pair $(d,f)$ correpsponds to a pair $(c, \tau)$ where $\tau: \Delta c \rightarrow F$ is a natural transformation i.e. a morphism in $\text{Funct}(J,C)$
• the moprhism $f'$ corresponds to a morphism $t: c \rightarrow r$ which is an arrow in $C$

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On why $(r, \nu)$ is terminal:

In the text, McLane writes $\lim F$ instead of $r$. I'll continue with $r$. Firstly, note that $\nu: \Delta r \rightarrow F$ is a natural transformation, and since the image of $\Delta c$ only has one object the components of $\nu$ can be indexed by the elements in the image of $F$ i.e. for every $F(i)$ where $i \in J$ we have one morphism $\nu_i: \Delta c \rightarrow F(i)$.

Thus being a terminal object in this category means that for any other $(e,\tau)$ where $e \in C$ and $\tau: \Delta e \rightarrow F$ we have that there is a unique $t: e \rightarrow r$ s.t. $ \tau= \nu \circ \Delta t$. Here $\Delta t$ is a morphism in $\text{Funct}(J,C)$ i.e. a natural transformation $\Delta e \rightarrow \Delta r$. However, since $\Delta$ is constant, this natural transformation only has a single component. This is why in the text, McLane simply writes all of the above condition as $\tau_i = \nu_i t$.