Suppose we have an open set $U$ in the plane and two $\cal C^\infty$ paths $\gamma,\eta:[a,b]\to U$ with the same endpoints (i.e., $P:=\gamma(a)=\eta(a)$ and $Q:=\gamma(b)=\eta(b)$). We say that $\gamma$ and $\eta$ are smoothly homotopic with fixed endpoints if there is a $\cal C^\infty$ function $H:[a,b]\times[0,1]\to U$ such that $H(a,s)=P$ for all $s$, $H(b,s)=Q$ for all $s$, $H(t,0)=\gamma(t)$ for all $t$ and $H(t,1)=\eta(t)$ for all $t$.
I have problems in showing that being smoothly homotopic is an equivalence relation. The problem is only in showing that it is a transitive relation. Can someone help me?
Let $H,F$ be two homotopies. Then concatenate them by defining $H\star F:[a,b]\times [0,2]\rightarrow U$ $$ H\star F =\begin{cases} F(x,t) \qquad &0\leq t \leq 1\\ H(x,t-1) \qquad &1\leq t\leq 2 \end{cases} $$ This is continuous but not necessarily smooth. Moreover the domain is not $[a,b]\times [0,1]$. Now the trick is to redefine this to a homotopy $H\diamond F:[a,b]\times[0,1]\rightarrow U$. You can do this by applying a smooth map $[0,2]\rightarrow [0,1]$ in the second coordinate which is constant around $1$.