Lemma 96.3.3. Let $f, g : \mathcal{X} \to \mathcal{Y}$ be $1$-morphisms of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $t : f \to g$ be a $2$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assigned to $t$ there are canonical isomorphisms of functors
\[ t^ p : g^ p \longrightarrow f^ p \quad \text{and}\quad {}_ pt : {}_ pf \longrightarrow {}_ pg \]
which compatible with adjointness of $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ and with vertical and horizontal composition of $2$-morphisms.
Proof.
Let $\mathcal{G}$ be a presheaf on $\mathcal{Y}$. Then $t^ p : g^ p\mathcal{G} \to f^ p\mathcal{G}$ is given by the family of maps
\[ g^ p\mathcal{G}(x) = \mathcal{G}(g(x)) \xrightarrow {\mathcal{G}(t_ x)} \mathcal{G}(f(x)) = f^ p\mathcal{G}(x) \]
parametrized by $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$. This makes sense as $t_ x : f(x) \to g(x)$ and $\mathcal{G}$ is a contravariant functor. We omit the verification that this is compatible with restriction mappings.
To define the transformation ${}_ pt$ for $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ define ${}_ y^ f\mathcal{I}$, resp. ${}_ y^ g\mathcal{I}$ to be the category of pairs $(x, \psi : f(x) \to y)$, resp. $(x, \psi : g(x) \to y)$, see Sites, Section 7.19. Note that $t$ defines a functor ${}_ yt : {}_ y^ g\mathcal{I} \to {}_ y^ f\mathcal{I}$ given by the rule
\[ (x, g(x) \to y) \longmapsto (x, f(x) \xrightarrow {t_ x} g(x) \to y). \]
Note that for $\mathcal{F}$ a presheaf on $\mathcal{X}$ the composition of ${}_ yt$ with $\mathcal{F} : {}_ y^ f\mathcal{I}^{opp} \to \textit{Sets}$, $(x, f(x) \to y) \mapsto \mathcal{F}(x)$ is equal to $\mathcal{F} : {}_ y^ g\mathcal{I}^{opp} \to \textit{Sets}$. Hence by Categories, Lemma 4.14.9 we get for every $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y})$ a canonical map
\[ ({}_ pf\mathcal{F})(y) = \mathop{\mathrm{lim}}\nolimits _{{}_ y^ f\mathcal{I}} \mathcal{F} \longrightarrow \mathop{\mathrm{lim}}\nolimits _{{}_ y^ g\mathcal{I}} \mathcal{F} = ({}_ pg\mathcal{F})(y) \]
We omit the verification that this is compatible with restriction mappings. An alternative to this direct construction is to define ${}_ pt$ as the unique map compatible with the adjointness properties of the pairs $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ (see below). This also has the advantage that one does not need to prove the compatibility.
Compatibility with adjointness of $(f^ p, {}_ pf)$ and $(g^ p, {}_ pg)$ means that given presheaves $\mathcal{G}$ and $\mathcal{F}$ as above we have a commutative diagram
\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{X})}(f^ p\mathcal{G}, \mathcal{F}) \ar@{=}[r] \ar[d]_{- \circ t^ p} & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_ pf\mathcal{F}) \ar[d]^{{}_ pt \circ -} \\ \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{X})}(g^ p\mathcal{G}, \mathcal{F}) \ar@{=}[r] & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{Y})}(\mathcal{G}, {}_ pg\mathcal{F}) } \]
Proof omitted. Hint: Work through the proof of Sites, Lemma 7.19.2 and observe the compatibility from the explicit description of the horizontal and vertical maps in the diagram.
We omit the verification that this is compatible with vertical and horizontal compositions. Hint: The proof of this for $t^ p$ is straightforward and one can conclude that this holds for the ${}_ pt$ maps using compatibility with adjointness.
$\square$
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