Identity as a Function From the Terminal Object to $Hom(x,x)$

Question

I saw the following definition of the identity here at 1:55. If $\mathcal{C}$ is a small category, $\forall x\in \mathcal{C}$, the identity can be thought of as a map from the one element set $\{*\}$ which is the terminal object to the endomorphisms of $x$. I.e. a function $f:\{*\}\to Hom(x,x)$ will pick out the identity $1_x$. I'm trying to make sense of this, here is my attempt.

So, since $\{*\}$ is terminal a map from $\{*\}\to \{*\}$ exists and is the identity. Similarly, a map $g:a\in Hom(x,x)\to x$ exists. Hence $g\circ f:\{*\}\to a\to \{*\}$ composes to the identity which can only happen if $f$ maps $\{*\}\mapsto 1_x=a$. Otherwise, $f$ would map $\{*\}$ to the zero object which can't happen.

Answer 1

Respectfully, it seems to me that you are confused. Certainly, if you have a one-element set $\{\ast\}$, then for any set $S$ and any element $s\in S$, it can be useful to consider $s$ as being the unique map $\{\ast\}\to S$ with the property that $\ast\mapsto s$. For any category $\mathcal{C}$ and any object $x\in\mathcal{C}$, we have a set $\mathrm{Hom}(x,x)$, and that set has an element $\mathbf{1}_x\in\mathrm{Hom}(x,x)$, so you can do what I just described. However:

• This has nothing to do with terminal objects in the category $\mathcal{C}$. (While it's true that a one-element set $\{\ast\}$ is a terminal object in the category $\mathsf{Set}$, there's no reason for one to also be an object of the category $\mathcal{C}$, much less a terminal object.)

• You can do the same thing for any other element $f\in\mathrm{Hom}(x,x)$, it's not special to the identity map $\mathbf{1}_x$.

Answer 2

Just note that you are working in the category Set, so the morphism $f$ is just a "usual" function, whose domain is a singleton, which $f$ sends to $id:x\to x$ in $hom (x,x)$. In fact, for each $\phi:x\to x$ there is an $f_{\phi}$ such that $f_{\phi}(*)=\phi$. This idea is more general: to each $s\in S\in Set$, there is an $f_s:\left \{ * \right \}\to S$ defined by $f_s(*)=s.$ Thus, elements of sets can be regarded as morphisms in Set. This is convenient, because in category theory it is the morphisms that give the most information about a given category.