I'm interested in the following problem, similar in vein to this other question. To put it simply, I have an adjoint pair $F\dashv G$ between categories $\mathrm{C}$ and $\mathrm{D}$ and I suppose that $\mathrm{D}$ is equipped with a symmetric monoidal category structure $(\otimes, U)$ (alternatively, you can add any reasonable adjective here to this structure).
Then, using the adjoint pair I can "transport" this monoidal structure on $\mathrm{D}$ to at least a bifunctor $\boxtimes:\mathrm{C}\times \mathrm{C}\to \mathrm{C}$ simply as:
$A\boxtimes B:= G(F(A)\otimes F(B))$
for any $A,B\in \mathrm{C}$.
Suppose there is a unit object $V$ for this bifunctor, in the sense that $A\boxtimes V\cong A$ for any object $A\in \mathrm{C}$. And say that an object $X$ is $\boxtimes$-invertible if there exists $X^{-1}$ such that $X\boxtimes X^{-1}\cong V$.
Question: Are there reasonable conditions (other than $F,G$ being inverse of each other) I can ask of $F,G, C,D,\otimes, U$ or $V$ so that if $X$ is an invertible object, I can conclude that $X\boxtimes\_:\mathrm{C}\to \mathrm{C}$ is an equivalence?
The usual proof does not work as we can't really expect any sort of associativity to work here.
This setting is a general one, but I am concretely interested in the particular case of a pair adjoint of derived functors $f^{\ast}\dashv f_{\ast}$ between derived categories $D^{b}(X)$, $D^{b}(Y)$ for smooth proj varieties but where the tensor structure in $D^{b}(Y)$ need not be $\otimes_{Y}^{\mathbb{L}}$.
