My question is prompted by 57589
If $X$ is an object in a monoidal category with unit $I$ then $Y$ is a left dual if we have $I\rightarrow Y\otimes X$ and $X\otimes Y\rightarrow I$ which satisfy the well-known zig-zag identities.
My question is: what is the homotopy version of this story? This is supposed to be equivalent to: if $X$, $Y$ and $C$ are complexes such that $Y$ is left dual to $X$ and $C$ is homotopy equivalent to $Y$ then what structure does $C$ have?
This example is evidently not an operad but it is a PROP. So, a subsidiary question is whether there is a theory of quadratic PROPS and Koszul PROPS which includes this example?
Another subsidiary question is whether the Spanier-Whitehead dual of a space is a dual in this sense. Note that the Spanier-Whitehead dual of a space is only defined up to stable homotopy.