The Stacks project

Lemma 7.44.2. Suppose the functor $u : \mathcal{C} \to \mathcal{D}$ satisfies the hypotheses of Proposition 7.14.7, and hence gives rise to a morphism of sites $f : \mathcal{D} \to \mathcal{C}$. In this case the pullback functor $f^{-1}$ (resp. $u_ p$) and the pushforward functor $f_*$ (resp. $u^ p$) extend to an adjoint pair of functors on the categories of sheaves (resp. presheaves) of algebraic structures. Moreover, these functors commute with taking the underlying sheaf (resp. presheaf) of sets.

Proof. We have defined $f_* = u^ p$ above. In the course of the proof of Proposition 7.14.7 we saw that all the colimits used to define $u_ p$ are filtered under the assumptions of the proposition. Hence we conclude from the definition of a type of algebraic structure that we may define $u_ p$ by exactly the same colimits as a functor on presheaves of algebraic structures. Adjointness of $u_ p$ and $u^ p$ is proved in exactly the same way as the proof of Lemma 7.5.4. The discussion of sheafification of presheaves of algebraic structures above then implies that we may define $f^{-1}(\mathcal{F}) = (u_ p\mathcal{F})^\# $. $\square$

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