Let $(E^{\cdot},d_E^{\cdot})$ be a cochain complex of complex vector bundles on a smooth compact manifold $X$. Now for each $E^i$ we could assign a connection $\nabla_E^i$ and obtain its curvature $(\nabla_E^i)^2:E^i\to E^i\otimes\Omega^2$. Then we define the chern character of $(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})$ to be $$ ch(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})：=str(\exp(-(\nabla_E^{\cdot})^2)) $$ where $str$ denotes the supertrace. We notice that the expression of $ch(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})$ has nothing to do with the differential $d_E^{\cdot}$ of the cochain complex.
Now consider two cochain complexes of complex vector bundles with connections $(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})$ and $(F^{\cdot},d_F^{\cdot},\nabla_F^{\cdot})$. Let $\phi: (E^{\cdot},d_E^{\cdot})\overset{\sim}{\to}(F^{\cdot},d_F^{\cdot})$ be a quasi-isomorphism between them. Then it is clear that $ch(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})$ and $ch(F^{\cdot},d_F^{\cdot},\nabla_F^{\cdot})$ should be cohomologous in $H^\cdot(X)$.
My question is: could we find the transgression formula in this case, i.e. could we construction a form $H$ which depends on $(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})$, $(F^{\cdot},d_F^{\cdot},\nabla_F^{\cdot})$ and $\phi$ such that $$ ch(E^{\cdot},d_E^{\cdot},\nabla_E^{\cdot})-ch(F^{\cdot},d_F^{\cdot},\nabla_F^{\cdot})=dH? $$
