Treated as functions from $S^1 \to Y$, $\omega$ and $\tau \circ \omega \circ {\tau}^{-1}$ are homotopic. How to prove this?

Question

Suppose that $\omega$ is a loop at $y_1 \in Y$ and $\tau : I \to Y$ is a path joining $y_0$ and $y_1, y_0 \neq y_1$. Show that there is a homotopy $H : I \times I \to Y$ from $\alpha$ to $\tau \circ \omega \circ {\tau}^{-1}$ such that $H(0, s) = \tau (s) = H(1, s)$.