Functoriality and change of basepoint for the fundamental group

Let \(h:(X,x_0)\to(Y,y_0)\) be continuous. Prove that the induced map \[ h_*:\pi_1(X,x_0)\to\pi_1(Y,y_0) \] is a well-defined group homomorphism.

For composable pointed maps \(h\) and \(k\), prove \[ (k\circ h)_*=k_*\circ h_*. \]

Prove that the identity map induces the identity homomorphism on \(\pi_1\).

Let \(f:X\to Y\) be continuous, let \(x_0,x_1\in X\), and let \(\alpha\) be a path from \(x_0\) to \(x_1\). If \(\beta_\alpha\) denotes change of basepoint along \(\alpha\), prove \[ \beta_{f\circ\alpha}\circ f_{*,x_0} =f_{*,x_1}\circ\beta_\alpha. \]