It's somewhat weird to see someone immediately jump to category theory when it's more likely to be about basic linear algebra.

That said convolution and correlation are not linear operators but bilinear operators, so I'm struggling a bit to see in what sense they're adjoint.

Maybe something like:

<Conv(x,y), z> = <y, Corr(x, z)>

which does actually work if you let the inner product be the integral of the product and the correlation the time-correlation (so the inner product is the correlation at time 0).