I'm reading Yang-Mills connections and Einstein-Hermitian metrics by Itoh and Nakajima. On definition 1.8 they define a notion for an Einstein-Hermitian connection $A$ by $$K_A = \lambda(E)\text{id}.$$

After this, they state

Using the Chern-Weil formula, we see that the constant $\lambda(E)$ is given by $$\lambda(E) = \frac{2\pi}{r(m-1)!\text{vol}(X)}\int_X c_1(E) \wedge \omega^{m-1}.$$

I have read quite a bit about Chern-Weil theory, but I have never met the so-called "Chern-Weil formula". They do not define it in the notes and I could not find a definition for this online. This certainly isn't the Chern-Weil homomorphism, but something else instead. Anyone here happen to know what is meant by this?