Lemma 7.19.1. There is a canonical map {}_ pu\mathcal{F}(u(U)) \to \mathcal{F}(U), which is compatible with restriction maps.
7.19 More functoriality of presheaves
In this section we revisit the material of Section 7.5. Let u : \mathcal{C} \to \mathcal{D} be a functor between categories. Recall that
u^ p : \textit{PSh}(\mathcal{D}) \longrightarrow \textit{PSh}(\mathcal{C})
is the functor that associates to \mathcal{G} on \mathcal{D} the presheaf u^ p\mathcal{G} = \mathcal{G} \circ u. It turns out that this functor not only has a left adjoint (namely u_ p) but also a right adjoint.
Namely, for any V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D}) we define a category {}_ V\mathcal{I} = {}_ V^ u\mathcal{I}. Its objects are pairs (U, \psi : u(U) \to V). Note that the arrow is in the opposite direction from the arrow we used in defining the category \mathcal{I}_ V^ u in Section 7.5. A morphism (U, \psi ) \to (U', \psi ') is given by a morphism \alpha : U \to U' such that \psi = \psi ' \circ u(\alpha ). In addition, given any presheaf of sets \mathcal{F} on \mathcal{C} we introduce the functor {}_ V\mathcal{F} : {}_ V\mathcal{I}^{opp} \to \textit{Sets}, which is defined by the rule {}_ V\mathcal{F}(U, \psi ) = \mathcal{F}(U). We define
{}_ pu(\mathcal{F})(V) := \mathop{\mathrm{lim}}\nolimits _{{}_ V\mathcal{I}^{opp}} {}_ V\mathcal{F}
As a limit there are projection maps c(\psi ) : {}_ pu(\mathcal{F})(V) \to \mathcal{F}(U) for every object (U, \psi ) of {}_ V\mathcal{I}. In fact,
{}_ pu(\mathcal{F})(V) = \left\{ \begin{matrix} \text{collections } s_{(U, \psi )} \in \mathcal{F}(U)
\\ \forall \beta : (U_1, \psi _1) \to (U_2, \psi _2) \text{ in }{}_ V\mathcal{I}
\\ \text{ we have } \beta ^*s_{(U_2, \psi _2)} = s_{(U_1, \psi _1)}
\end{matrix} \right\}
where the correspondence is given by s \mapsto s_{(U, \psi )} = c(\psi )(s). We leave it to the reader to define the restriction mappings {}_ pu(\mathcal{F})(V) \to {}_ pu(\mathcal{F})(V') associated to any morphism V' \to V of \mathcal{D}. The resulting presheaf will be denoted {}_ pu\mathcal{F}.
Proof. This is just the projection map c(\text{id}_{u(U)}) above. \square
Note that any map of presheaves \mathcal{F} \to \mathcal{F}' gives rise to compatible systems of maps between functors {}_ V\mathcal{F} \to {}_ V\mathcal{F}', and hence to a map of presheaves {}_ pu\mathcal{F} \to {}_ pu\mathcal{F}'. In other words, we have defined a functor
{}_ pu : \textit{PSh}(\mathcal{C}) \longrightarrow \textit{PSh}(\mathcal{D})
Lemma 7.19.2. The functor {}_ pu is a right adjoint to the functor u^ p. In other words the formula
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ p\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(\mathcal{G}, {}_ pu\mathcal{F})
holds bifunctorially in \mathcal{F} and \mathcal{G}.
Proof. This is proved in exactly the same way as the proof of Lemma 7.5.4. We note that the map u^ p{}_ pu \mathcal{F} \to \mathcal{F} from Lemma 7.19.1 is the map that is used to go from the right to the left.
Alternately, think of a presheaf of sets \mathcal{F} on \mathcal{C} as a presheaf \mathcal{F}' on \mathcal{C}^{opp} with values in \textit{Sets}^{opp}, and similarly on \mathcal{D}. Check that ({}_ pu \mathcal{F})' = u_ p(\mathcal{F}'), and that (u^ p\mathcal{G})' = u^ p(\mathcal{G}'). By Remark 7.5.5 we have the adjointness of u_ p and u^ p for presheaves with values in \textit{Sets}^{opp}. The result then follows formally from this. \square
Thus given a functor u : \mathcal{C} \to \mathcal{D} of categories we obtain a sequence of functors
u_ p, u^ p, {}_ pu
between categories of presheaves where in each consecutive pair the first is left adjoint to the second.
Lemma 7.19.3. Let u : \mathcal{C} \to \mathcal{D} and v : \mathcal{D} \to \mathcal{C} be functors of categories. Assume that v is right adjoint to u. Then we have
u^ ph_ V = h_{v(V)} for any V in \mathcal{D},
the category \mathcal{I}^ v_ U has an initial object,
the category {}_ V^ u\mathcal{I} has a final object,
{}_ pu = v^ p, and
u^ p = v_ p.
Proof. Proof of (1). Let V be an object of \mathcal{D}. We have u^ ph_ V = h_{v(V)} because u^ ph_ V(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(U), V) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(U, v(V)) by assumption.
Proof of (2). Let U be an object of \mathcal{C}. Let \eta : U \to v(u(U)) be the map adjoint to the map \text{id} : u(U) \to u(U). Then we claim (u(U), \eta ) is an initial object of \mathcal{I}_ U^ v. Namely, given an object (V, \phi : U \to v(V)) of \mathcal{I}_ U^ v the morphism \phi is adjoint to a map \psi : u(U) \to V which then defines a morphism (u(U), \eta ) \to (V, \phi ).
Proof of (3). Let V be an object of \mathcal{D}. Let \xi : u(v(V)) \to V be the map adjoint to the map \text{id} : v(V) \to v(V). Then we claim (v(V), \xi ) is a final object of {}_ V^ u\mathcal{I}. Namely, given an object (U, \psi : u(U) \to V) of {}_ V^ u\mathcal{I} the morphism \psi is adjoint to a map \phi : U \to v(V) which then defines a morphism (U, \psi ) \to (v(V), \xi ).
Hence for any presheaf \mathcal{F} on \mathcal{C} we have
\begin{eqnarray*} v^ p\mathcal{F}(V) & = & \mathcal{F}(v(V)) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{v(V)}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(u^ ph_ V, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{D})}(h_ V, {}_ pu\mathcal{F}) \\ & = & {}_ pu\mathcal{F}(V) \end{eqnarray*}
which proves part (4). Part (5) follows by the uniqueness of adjoint functors. \square
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