I am looking for a semi-computational reference for the following.

I have a morphism of categories f: C --> D, both small (finite even) categories. I then want to consider the categories of C-modules and D-modules in abelian groups (functors from these categories into abelian groups). Is there any spectral sequence or relation between two notions of Tor (with respect to the day convolution tensor product)?

C and D symmetric monoidal?

and f?

I am thinking a bit more carefully of the example I have in mind.

Lets assume thatf is symmetric monoidal.

In my example I know for sure that D is.

f is a sort of collaps/fold functor.

if f is symmetric monoidal then so is the map Psh(C)-->Psh(D), where we take presheaves of complexes. so f_!(a \otimes b)= f_! a \otimes f_!b. In terms of Tor stuff, maybe it's better to write as `$Lf_!(a\otimes^{\mathbb{L}} b) = R=Lf_!(a) \otimes^{\mathbb{L}} Lf_!(b)$. So you can compute that with a composite functor sseq, right?

What's a good intuition for Lurie's definition of a "cherent $\infty$-operad"?

It’s supposed to be the condition you need in order to ‘compose multilinear maps’ between operadic modules