Hi everyone

I have this diagram, which I must show commutes, i.e. that

$\displaystyle \hat{\beta}\circ (h_{x_0})_*=(h_{x_1})_*\circ \hat{\alpha}$

Where $\displaystyle \alpha$ is a path in X and $\displaystyle \beta=h\circ\alpha$

and $\displaystyle h:X->Y$ is continous with $\displaystyle h(x_0)=y_0$ and $\displaystyle h(x_1)=y_1$

But I'm a little stuck. What I do is this:

Is that correct? But I'm not sure if it's allowed to just say $\displaystyle \overline{h\circ\alpha}=h\circ \bar{\alpha}$. What is the argument for this, that h is continous?