ct.category theory - What tensor product of chain complexes satisfies the usual universal property?

Recall that a chain complex is a (finite) diagram of the form

$$V = { dots to V_3 overset{d_3}to V_2 overset{d_2}to V_1 overset{d_1}to V_0 to 0 }$$
where the $V_n$ are (finite-dimensional) vector spaces and for each $n$, $d_n circ d_{n+1} = 0$. If $V$ and $W$ are chain complexes, a chain map $f: V to W$ is a map $f_n : V_n to W_n$ for each $n$ such that all the obvious squares commute — "$[d,f]=0$" — and the pair (chain complexes, chain maps) defines a category. In fact, it is a 2-category: the 2-morphisms between $f,g : V rightrightarrows W$ are the chain homotopies, i.e. a system of maps $h_n: V_n to W_{n+1}$ such that "$[d,h] = f-g$". The category of chain complexes has a biproduct (both a product and a coproduct) $oplus$ given by the pointwise direct sum.

I thought I knew what the tensor product of chain complexes was. Namely, if $V$ and $W$ are chains, then the usual thing is to define

$$(Votimes W)_n = bigoplus_{k=0}^n V_k otimes W_{n-k}$$
and the chain maps are the sums of the obvious tensor products of differentials, decorated with signs.

But now I'm not sure why this is the tensor product picked. Namely, if I have a linear category, I think that a tensor product $V otimes W$ should satisfy the following universal property: for any $X$, $hom(V otimes W,X)$ should be naturally isomorphic to the space of bilinear maps $V times W to X$. Now, I've never really known how to write down the word "bilinear" in a general category, without refering to individual points. But I think I do know what the "set" $V times W$ is when $V$ and $W$ are chains — it's the set underlying $V oplus W$ — and then I think I do know what bilinear maps should be.

In any case, then it's clear that the usual tensor product is not this. For example, if $V,W$ have no non-zero terms above degree $n$, then the bilinear maps $V times W to X$ I think cannot be interesting above degree $n$, whereas the above $otimes$ has terms in degree $2n$.

In any case, in HDA6, Baez and Crans consider two-term chain complexes $V_1 to V_0$ (they argue that these are the same as "2-vector-spaces"), and then construct a different tensor product, given by:

$$Votimes W = { (V_1 otimes W_1) oplus (V_1 otimes W_0) oplus (V_0otimes W_1) to (V_0 otimes W_0) }$$
where the differential is the sum of the obvious tensor products of differentials and identity maps. They then assert that this tensor product satisfies the correct universal property, although they leave the details to the reader.

This leads naturally to:

Question: What is the precise universal property that $otimes$ ought to have, and what "product" of chain maps satisfies this universal property?

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