L&S Cat Theory: Attempted solutions to "Conceptual Mathematics" by Lawvere and Schanuel

I don’t know if this will become relevant, but I’ve got this overwhelming urge to complete the diagram above this problem:

My intuition suggests that the existence of a retraction like $r$ in the diagram depends whether or not $Y\subset X\text{.}$ I think that’s conceptually related to the “dual” of this exercise.

I think there’s also an implication that the result I’m looking for follows from the “universal property of products”. This was defined over a (possibly empty) set of indices $I$ where each $i$ in $I$ defines a (possibly duplicated) object $C_i$ in the category $\mathcal{C}\text{.}$

In this hypothetical category $\mathcal{C}$ with initial object $S$ and terminal object $T\text{,}$ there should be unique maps $\alpha ,\beta$ where $S\stackrel{\alpha }{\to }X\stackrel{f}{\to }Y\stackrel{\beta }{\to }T$ satisfy the associative property. Let’s start enumerating these objects $C_i$ of our category: $C_0=S\text{,}$ $C_1=X\text{,}$ $C_2=Y\text{,}$ $C_3=T\text{.}$

By the universalized definition of product, for any $X$ there should be a unique object $P$ and map $X\stackrel{f}{\to }P\text{,}$ and maps $P\stackrel{p_i}{\to }{C}_i$ for each $i$ such that $p_{i}f=f_i$ for every $X\stackrel{f_i}{\to }{C}_I\text{.}$ Since this product is also an object in the category, let’s call it $C_4=S\times X\times Y\times T\text{.}$ What if we just need to iterate over all of these $C_i$ to handle the cases where $X=Y$ and $X\ne Y$ at the same time?

Let’s start with $X=C_0=S\text{.}$ Any map $X\stackrel{f_0}{\to }{C}_0$ would have to the unique map $S\stackrel{1_S}{\to }S\text{.}$ Likewise, any $X\stackrel{f_1}{\to }{C}_1$ would be the unique map $S\to X\text{,}$ $X\stackrel{f_2}{\to }{C}_2$ would be the unique map $S\to Y\text{,}$ $X\stackrel{f_3}{\to }{C}_3$ would be the unique map $S\to T\text{,}$ $X\stackrel{f_4}{\to }{C}_4$ would be the unique map $S\to X\times Y\text{.}$

Let’s move on $X=C_1=X\text{.}$ If any map $X\stackrel{f_0}{\to }{C}_0$ existed, that would imply that $X$ is itself an intial object. We could then used the uniqueness of the inverse map $C_1\to C_0$ to uniquely define the rest of the maps we’re looking for through precomposition with the maps we saw previously. In fact, I think we can go one step further and assume that no $C_i\to C_0$ exists for any $i>0\text{.}$

Assuming that no map $X\to C_0$ exists, then it seems reasonable to conclude that the only way $P\stackrel{p_1}{\to }X$ can satisfy $p_{1}p=f_1$ for all $X\stackrel{f_1}{\to }X$ is if $p_{1}p=1_X$ for all $X\text{.}$ I think this partially explains the wildcard notation of $?$ in the definition of $\mathrm{\Gamma }=⟨?,f⟩\text{.}$

The existence of some $X\stackrel{f}{\to }Y$ necessarily means we have maps $T\stackrel{x_i}{\to }X\stackrel{f}{\to }Y\stackrel{\overline{y_i}}{\to }T\text{.}$ I’m thinking we could have take the product $X\to X\times X$ that maps $x_i$ to the pair $⟨x_i,x_i⟩\text{.}$ and then somehow compare this with the projection map $X\to X\times Y$ that pairs each $x_i$ with the respective $y_i=f{x}_i\text{.}$ I’m thinking we can do this by “stepping out” somehow through a shared map $X\to X\times (X+Y)\text{.}$