I'm reading [Yang–Mills connections and Einstein–Hermitian metrics  by Itoh and Nakajima](https://doi.org/10.2969/aspm/01820395). On definition 1.8 they define a notion for an Einstein–Hermitian connection $A$ by $$K_A = \lambda(E)\mathrm{id}.$$

After this, they state 

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

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. Does anyone here happen to know what is meant by this?