I am reading the book on 2D topological quantum field theories by Joachim Kock and I am confused with the example on gluing two smooth 1-manifolds. It is on page 38-39 of the book.
Take two smooth 1-manifolds, $M_0$ and $M_1$. They have common boundary $\Sigma = \{p\}$.
Fix a chart on $M_0$ near $p$, say $f_0 : U_0 \rightarrow \mathbb{R}_- = \{x\in \mathbb{R} : x \le 0\}$. Pick a chart on $M_1$ near $p$, say $f_1:U_1 \rightarrow \mathbb{R}_+ = \{x\in \mathbb{R} : x \ge 0\}$. We can glue these two charts to give $f$ and define $U = U_0 \sqcup_\Sigma U_1$. So we have somewhat glued the two manifolds.
We can get another chart $g$ on $U$ by defining $g := f \alpha$, where $\alpha$ is a transition function \begin{align*} \alpha &: \mathbb{R} \rightarrow \mathbb{R} \\ & x \mapsto \begin{cases} x ~~~~ \text{for $x\le 0$}\\ x^2 ~~ \text{for $x\ge 0$}. \end{cases} \end{align*}
$\alpha$ is obviously not differentiable (at $x=0$) and hence we have two smooth structures on $U$, $(U,f)$ and $(U,g)$.
Now we want to show that these two smooth structures are diffeomorphic and the diffeomorphism is not unique.
An obvious diffeomorphism is $fg^{-1}$, i.e. $fg^{-1}$ is smooth because $f^{-1} (fg^{-1}) g$ is the identity, and $(fg^{-1})^{-1}$ is also smooth for the same reason. But $fg^{-1}$ is not the identity map from $U$ to $U$.
Here comes the part where I am confused. The book went on to say "this diffeomorphism is not compatible with the inclusions $U_0 \rightarrow U \leftarrow U_1$, so there is no universal property." I don't quite understand what it means, and how it leads to the conclusion that the diffeomorphism is not unique.