Remark 7.15.4. (Set theoretical issues related to morphisms of topoi. Skip on a first reading.) A morphism of topoi as defined above is not a set but a class. In other words it is given by a mathematical formula rather than a mathematical object. Although we may contemplate the collection of all morphisms between two given topoi, it is not a good idea to introduce it as a mathematical object. On the other hand, suppose $\mathcal{C}$ and $\mathcal{D}$ are given sites. Consider a functor $\Phi : \mathcal{C} \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$. Such a thing is a set, in other words, it is a mathematical object. We may, in succession, ask the following questions on $\Phi $.

Is it true, given a sheaf $\mathcal{F}$ on $\mathcal{D}$, that the rule $U \mapsto \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\Phi (U), \mathcal{F})$ defines a sheaf on $\mathcal{C}$? If so, this defines a functor $\Phi _* : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$.

Is it true that $\Phi _*$ has a left adjoint? If so, write $\Phi ^{-1}$ for this left adjoint.

Is it true that $\Phi ^{-1}$ is exact?

If the last question still has the answer “yes”, then we obtain a morphism of topoi $(\Phi _*, \Phi ^{-1})$. Moreover, given any morphism of topoi $(f_*, f^{-1})$ we may set $\Phi (U) = f^{-1}(h_ U^\# )$ and obtain a functor $\Phi $ as above with $f_* \cong \Phi _*$ and $f^{-1} \cong \Phi ^{-1}$ (compatible with adjoint property). The upshot is that by working with the collection of $\Phi $ instead of morphisms of topoi, we (a) replaced the notion of a morphism of topoi by a mathematical object, and (b) the collection of $\Phi $ forms a class (and not a collection of classes). Of course, more can be said, for example one can work out more precisely the significance of conditions (2) and (3) above; we do this in the case of points of topoi in Section 7.32.

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