Let $k$ be a commutative ring, and let $V$ be a complex of $k$-modules (more in general, we can take an $\mathcal A$-dg-module, where $\mathcal A$ is a dg-category. We can construct the bar resolution $B(V)$ of $V$, which comes with a quasi-isomorphism $B(V) \to V$ (here is a related post). My question is the following:
Does $B(-)$ define a dg-functor from the dg-category of complexes to itself? Do the maps $B(V) \to V$ define a natural transformation of dg-functors?
I think that $B$ should at least be a (ordinary) functor from the (model) category of complexes to itself. It would be very nice if it were actually dg-functorial. Or, perhaps, is it $A_\infty$-functorial?