Morphisms of localizations of topoi Let $T$ be a topos, and $f : F\rightarrow G$ be a morphism in $T$. Exercise 2.F of Olsson's book Algebraic spaces and stacks asks us to show that there is a morphism
$$(f^*,f_*) : T/F\rightarrow T/G$$
of topoi where $f^*$ sends $(H\rightarrow G)$ to $H\times_G F\rightarrow F$, and $f_*$ is characterized by the property that for any $H\rightarrow G$ in $T/G$ and $M\rightarrow F$ in $T/F$ we have
$$Hom_{T/G}(H,f_*M) = Hom_{T/F}(H\times_G F,M)$$
My first thought was to define $f_*(M\rightarrow F)$ as the composite $M\rightarrow F\stackrel{f}{\rightarrow}G$. Defined this way, any morphism $H\rightarrow f_*M$ in $T/G$ naturally gives a morphism $H\times_G F\rightarrow M$ in $T/F$, but the converse doesn't seem to be obvious.
My question is - how should we define $f_*$?
One could try to define $f_*M$ as a presheaf $T/F\rightarrow\textbf{Sets}$ given by $(f_*M)(H) := Hom_{T/F}(H\times_G F,M)$, but then it's unclear if $f_*M$ defined this way is represented by an object of $T/F$.
 A: Hint: Consider the case $G=\ast$ (the terminal object) first and look at the larger set $$\text{Hom}_{{\mathscr T}}(H\times F, M)\supset\text{Hom}_{{\mathscr T}/F}(H\times F, M).$$ Since any topos is cartesian closed, we have $\text{Hom}_{{\mathscr T}}(H\times F, M)\cong\text{Hom}_{\mathscr T}(H,M^F)$. Can you work out what subobject of $M^F$ to restrict that correspondence to $\text{Hom}_{{\mathscr T}/F}(H\times F, M)$? Note that $M$ comes with a structure morphism $M\to F$ we haven't considered so far.
In general, you can reduce to $G=\ast$ by noticing that ${\mathscr T}/F\cong ({\mathscr T}/G)/(F\to G)$.
Side remark: From the perspective of categorical logic, in case of $G=\ast$ the adjunction corresponds to the equivalence of $x:F\vdash P\Rightarrow Q(x)$ and $\emptyset\vdash P\Rightarrow \forall_x Q(x)$. 
Amendment: Of course the crucial part here is to know that any topos is cartesian closed (and that over-categories of topoi are again topoi). You may consult e.g. MacLane-Moerdijk's book Sheaves in Geometry and Logic for that, but in a nutshell the idea is constructing $M^F$ as the subobject of the power object ${\mathcal P}(F\times M)$ (which exists by definition of a topos) consisting of those $A\subset F\times M$ which satisfy $\forall x:F
,\ \exists! y:M,\ (x,y)\in A$. 
