I'm curious how to show how this matrix:

$$c = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$

is the Choi–Jamiołkowski matrix of a quantum channel for any $\lambda \in [0,1]$

Edit: Additional question: What is the Klaus representation (operator-sum representation) of the quantum channel T that is described by c and show that $T_{1/2}$ is an entanglement breaking quantum channel and describe its action on the bloch sphere.