Simple "Pointed Set" Question from Aluffi's Algebra $$\require{AMScd}:$$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.
 A: 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 $t=f_T(*)$.
A: 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 $\sigma(s)=t$.
$$\begin{array}{ccccc}
 &  & \star\\
 & \stackrel{f}{\swarrow} &  & \stackrel{g}{\searrow}\\
\left(S,s\right) &  & \stackrel{\sigma}{\longrightarrow} &  & \left(T,t\right)
\end{array}$$
