  Let $X\rightarrow S$ be a smooth projective surjective morphism of relative
dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let
$L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently
large positive integer $m$, there is a canonical isomorphism of the Deligne
pairing $\langle L ,\cdots , L\rangle\rightarrow S$ with the determinant line
bundle ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$
\cite{PRS}. If we fix a hermitian structure on $L$ and a relative K\"ahler form
on $X$, then each of the line bundles ${\rm Det}((L- {\mathcal O}_{X})^{\otimes
(n+1)}\otimes L^{\otimes m})$ and $\langle L\, ,\cdots\, ,L\rangle$ carries a
distinguished hermitian structure. We prove that the above mentioned
isomorphism between $\langle L\, ,\cdots\, ,L\rangle\longrightarrow S$ and
${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ is
compatible with these hermitian structures. This holds also for the isomorphism
in \cite{BSW} between a Deligne paring and a certain determinant line bundle.
