Reference for dualizable chain complexes

Question

Let $k$ be a commutative ring. Let $Ch(k)$ denote the monoidal category of chain complexes. I need a reference, including a proof, for the following "folklore fact" (see for example here after Definition 1.2.1):

A chain complex $A \in Ch(k)$ is dualizable if and only if it is bounded and each $A_n$ is a dualizable, i.e. finitely generated projective $k$-module.

The dual is then $(A^*)_n = (A_{-n})^*$.

Answer 1

This is Proposition 1.6 of 'Duality, Trace and Transfer' by Dold and Puppe, http://www.maths.ed.ac.uk/~aar/papers/doldpup2.pdf.