Simple "Pointed Set" Question from Aluffi's Algebra

Question

$$\require{AMScd}:$$Forgive me for this extremely basic question. I am going through Aluffi's textbook ``Algebra", and in an early example, he writes the following:

Let $C = Set$ and $A = $ a fixed singleton $\big\{*\big\}$. Call the resulting category $Set^{*}$. An object of $Set^{*}$ is a morphism $f:\big\{*\big\} \to S$ in $Set$, where $S$ is any set. We can denote an object of $Set^{*}$ as a pair $(S,s)$, where $S$ is any set and $s \in S$ is any element of $S$. A morphism between two such objects $(S,s) \to (T,t)$ corresponds to a set function $\sigma: S \to T$ such that $\sigma(s) = t$.

My question is: how can this be expressed diagrammatically? I.e. how can the morphism be defined diagrammatically? I am tempted to write the following, but my feeling is that would be incorrect.

\begin{CD} \big\{s\big\} @>{\sigma}>> \big\{t\big\}\\ @V{f}VV @VV{g}V \\ S @>{\sigma}>> T \end{CD}

Why is it incorrect? Well, this implies $g\sigma(s) = g(t) = T = \sigma( S )= \sigma f(s)$...which makes me uncomfortable.

I apologize if this question seems overly trivial.

Answer 1

It is not a stupid question at all. It is good to try to draw diagrams, but the best description of $\mathrm{Set}^*$ to do that is the first you gave, not the second. Indeed, you can draw a morphism $\sigma \colon S \to T$ as a commutative triangle: you have maps of sets

$$f_S \colon \{*\} \to S$$ $$f_T \colon \{*\} \to T$$ $$\sigma \colon S \to T.$$ The fact that the triangle commutes means exactly that $\sigma(s) = t$, where $s=f_S(*)$ and $t=f_T(*)$.

Answer 2

Here $S$ and $T$ are sets with $s\in S$ and $t\in T$.

Then $f:\star\to\langle S,s\rangle$ and $g:\star\to\langle T,t\rangle$ are objects in $\mathbf{Set}^{\star}$.

They are functions $\star\to S$ and $\star\to T$ respectively, prescribed by $f(\star)=s$ and $g(\star)=t$ respectively.

Arrows $\sigma:\langle S,s\rangle\to\langle T,t\rangle$ are functions $S\to T$ that make the following diagram commute by satisfying $\sigma(s)=t$.

$$\begin{array}{ccccc} & & \star\\ & \stackrel{f}{\swarrow} & & \stackrel{g}{\searrow}\\ \left(S,s\right) & & \stackrel{\sigma}{\longrightarrow} & & \left(T,t\right) \end{array}$$