Let $M$ be a compact manifold and $L_0,L_1$ two compact submanifolds. Let's definte $P(L_0,L_1):=\{\gamma :[0,1]\rightarrow M|$smooth $, \gamma(0)\in L_0, \gamma(1)\in L(1)\}$. Now fix $\gamma'\in P(L_0,L_1)$ and consider it's connected component which we denote by $P(\gamma')$, and relative to $P(\gamma')$ we can consider it's universal covering $\tilde P(\gamma')$, so that it's elements are of the form $[\gamma,w]$ where $w$ is a smooth path in $P(\gamma')$ between $\gamma'$ and $\gamma$. Now suppose we have in $\tilde P(\gamma')$ two representatives of the same class $[\gamma,w]$ and $[\gamma,w']$. Then why do we have the existece of a map $\bar w\# w':S^1\times [0,1]\rightarrow M$ such that $\bar w\# w'(s,i)$ is a loop in $L_i$ for $i=0,1$?
I know that if they represent the same class we have an homotopy between $w$ and $w'$ but this homotopy would give us a map $H:[0,1]\times [0,1]\times [0,1]\rightarrow M$ such that $H(0,0,.)=\gamma', H(1,0,.)=\gamma', H(1,1,.)=\gamma$ and $H(0,1,.)=\gamma$ and I am not sure how from this we get the desired map $\bar w\# w'$. Any help is appreciated, thanks in advance.