# Functoriality for presheaves

Let $$\mathcal{C}$$ be a small category. Functoriality for presheaves says that for any functor $$u\colon\mathcal{C}\to\mathcal{D}$$ the precomposition functor $$u^*\colon PSh(\mathcal{D})\to PSh(\mathcal {C})$$ has two adjoints $$u_!, u_*$$ on the left and on the right correspondingly, see SGA 4, Exposé I, Proposition 5.1 or ncatlab or stacks 1, stacks 2. It is a particular case of the Kan extension. (Here we consider only presheaves of sets.) These functors can be defined as follows: $$u_!F(Y)=\varinjlim_{Y\to u(X)} F(X)$$ $$u_*F(Y)=\varprojlim_{u(X)\to Y} F(X)$$ for any $$Y\in \mathcal{D},\; F\in PSh(\mathcal{C})$$, see Theorem 2.3.3 on Kan extensions in Kashiwara-Schapira "Categories and Sheaves" (here the limits are reversed since we work with contravariant functors = presheaves).

I try to understand this construction in the case of topological spaces. Let $$f\colon Y\to X$$ be a continuous map of topological spaces. Let $$Op(X)$$, $$Op(Y)$$ be the categories of open subsets where morphisms are inclusions. Then there is a functor $$u\colon Op(X)\to Op(Y)$$ which sends $$U\subset X$$ to $$f^{-1}U\subset Y$$. Then we get a functor $$u^*\colon PSh(Y)\to PSh(X)$$ which is the usual pushforward $$f_*$$. The functor $$u_!F(V)=\varinjlim_{V\subset f^{-1}(U)} F(U)=\varinjlim_{f(V)\subset U} F(U)=f^{-1}F(V)$$ is the usual pullback for presheaves. However, I cannot unfold the definition of the second adjoint $$u_*$$. By the general definition above, one should have $$u_*F(V)=\varprojlim_{f^{-1}(U)\subset V} F(U).$$ But then it is not clear how the restriction maps look like: for an inclusion $$V\subset V^\prime$$ of open subsets in $$Y$$ we should have a map $$u_*F(V^\prime)=\varprojlim_{f^{-1}(U)\subset V^\prime} F(U) \to \varprojlim_{f^{-1}(U)\subset V} F(U)=u_*F(V).$$ However, each $$f^{-1}(U)$$ contained in $$V$$ is contained in $$V^\prime$$ and not vice versa. So the map seems to be in the opposite direction $$\varprojlim_{f^{-1}(U)\subset V} F(U) \to \varprojlim_{f^{-1}(U)\subset V^\prime} F(U).$$ Thus, what we have constructed is not a presheaf of sets but rather a covarinat functor $${Op(Y)}\to Sets$$. I would be grateful if you could tell me where is my mistake.

In general, if $$J \subseteq I$$, and you have an inverse system of algebraic objects $$X_i$$, then you have a canonical map $$\varprojlim_{i\in I} X_i \to \varprojlim_{i\in J} X_i.$$ One way to see this is as a restriction of the "selection of a subset of coordinates" operation $$\prod_{i\in I} X_i \to \prod_{i\in J} X_j$$ (and verify that if you have the consistency condition for an element of $$\prod_{i\in I} X_i$$ to be in $$\varprojlim_{i\in I} X_i$$, then its image satisfies the consistency condition to be in $$\varprojlim_{i\in J} X_i$$). Another way is to construct it using the universal property of $$\varprojlim_{i\in J} X_i$$.
In your case, you have that $$\{ U \in \operatorname{Op}(X) \mid f^{-1}(U) \subseteq V \} \subseteq \{ U \in \operatorname{Op}(X) \mid f^{-1}(U) \subseteq V' \}$$.
Hint: The limit of a diagram $$D$$ has a canonical map to the limit of a subdiagram $$D'$$, induced from projection maps out of the limit of $$D$$, using the universal property of the limit of $$D'$$.