I'm having some problems at understanding how logical functors for elementary toposes work. I understood that logical functors preserve truth values of formulas in the internal logic. Now my question is:

Say that I have a logical functor between elementary toposes $L:\mathcal{E} \to \mathcal{F}$. Moreover, I can define an internal quantale $Q$ in $\mathcal{E}$ as an object $Q$ of $\mathcal{E}$ together with morphisms $\bigvee : PQ \to Q$ and $\otimes: Q \times Q \to Q$ such that their behaviour is defined in terms of the internal logic of $\mathcal{E}$; as an instance, I can require the following formula (among others, this is just an example) to hold: $$x \otimes (\bigvee_i y_i) = \bigvee_i (x \otimes y_i).$$ I can then call this object $Q$ (together with the required morphisms and formulas) an "internal quantale". My questions are:

1) If I apply the logical functor $L$ to $Q$, is $LQ$ together with morphisms $L(\bigvee), L(\otimes)$ an internal quantale in $\mathcal{F}$? My understanding should be "yes", but I'm struggling a bit to prove this: My problem is that I don't know precisely how to relate variables in $\mathcal{E}$ with variables in $\mathcal{F}$. I would be able to say stuff like $L(x \otimes y) = x' L(\otimes) y'$ where $x,y$ are variables of type $Q$ and $x',y'$ variables of type $LQ$, but I don't understand what relationship there is between $x, x'$ and $y, y'$.

2) Suppose I have two morphisms of $\mathcal{E}$, say $R:A \to Q$ and $S: B \to Q$. I can define the "product" $(R \square S): A \times B \to Q$ as $$R \square S(a,b) = (R(a) \otimes S(b)).$$

Given for granted that the previous point holds, I can define the same product on $\mathcal{F}$ replacing $\otimes$ (the multiplication of the quantale $Q$ in $\mathcal{E}$) with $L(\otimes)$ (the multiplication of the quantale $LQ$ in $\mathcal{F}$). Is it true in general that $L(R \square S) \circ \sigma = LR \square LS$ - here $\sigma$ is the canonical isomorphism $LA \times LB \to L(A \times B)$ - or do I need to make more assumptions on $L$? Again, my problem is that I don't understand how variables relate from a topos to another: I'd like to check what happens if I do $(L(R \square S) \circ \sigma) (a,b)$, where $a,b$ are variables of type $LA, LB$, respectively, but I don't know how to let these variables "slide" into $L$.

What am I getting wrong? Thanks for your time.


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