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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