Understanding pointed simplicial sets For pointed simplicial sets there are two equivalent definitions of the basepoint. Let $\Delta^0$ be the simplicial set with only one vertex in each degree. Let $X$ be a simplicial set. Then a basepoint in $X$ is either a simplicial map $\varphi:\Delta^0\to X$ or a distinguished point $\ast\in X_0$. I see that the map $\varphi$ specifies a point in $X_0$ (and any $X_i$ for $i>0$).
My question: How does, on the other hand, a point $\ast\in K_0$ determine a map $\varphi:\Delta^0\to X$? How are these two notions equivalent?
 A: $\newcommand{\De}{\Delta}$
$\newcommand{\Set}{\text{Set}}$
This is all very much representable functors and Yoneda lemma stuff.
A simplicial set $X$ is a contravariant functor from the simplex category
$\De$ to $\Set$. So for each object $[n]$ of $\De$ we have
a set $X_n$ and also appropriate maps between these. Simplicial sets
form a category $\hat\De=\Set^{\De^{\text{op}}}$. Yoneda's lemma shows that
there is an embedding $\De\to\hat\De$ where $[n]$
is mapped to the simplicial set $\De^n$ defined by $\De^n:[m]\mapsto
\text{Mor}_\De([m],[n])$. For each $n$ there is a functor
from $\hat\De$ to $\Set$ given by $X\mapsto X_n$.
Yoneda's lemma tells us this functor is represented by $\De^n$, that is
$X_n\cong \text{Map}_{\hat\De}(\De^n,X)$.
Your question about pointing is the case $n=0$ looked at in these
two ways. Picking a point in $X_0$ is picking a point in $X_0$, but that
is naturally equivalent to picking a simplicial map from $\De^0$
to $X$; this is the equivalence
$X_0\cong \text{Map}_{\hat\De}(\De^0,X)$.
