  In this note, we give a proof for a variant of the functorial
Deligne-Riemann-Roch theorem in positive characteristic based on ideas
appearing in Pink and R\"ossler's proof of the Adams-Riemann-Roch theorem in
positive characteristic (see \cite{Pi}). The method of their proof appearing in
\cite{Pi}, which is valid for any positive characteristic and which is
completely different from the classical proof, will allow us to prove the
functorial Deligne-Riemann-Roch theorem in a much easier and more direct way.
Our proof is also partially compatible with Mumford's isomorphism.
