Chern forms and tensor products

Question

Let $E\to X$ be a rank $r$ holomorphic vector bundle over a Kahler manifold and let $L\to X$ be a holomorphic line bundle. The following relation between Chern classes is (well) known: $$c_2(E\otimes L)=c_2(E)+(r-1)c_1(E)c_1(L)+\binom{r}{2}c_1(L)^2.$$ Assume now that $E,L$ are endowed with Hermitian metrics $h_E,h_L$. Does the above relation holds at the level of differential forms ? In other terms do we have the following identity of Chern forms $$c_2(E\otimes L,h_E\otimes h_L)=c_2(E,h_E)+(r-1)c_1(E,h_E)c_1(L,h_L)+\binom{r}{2}c_1(L,h_L)^2?$$