Categories and the direct image functor

I've just started learning about sheaves and yesterday I had a look at MacLane/Moerdijk's book "Sheaves in Geometry and Logic". I've discovered something in this book on which I'd like some clarification.

Suppose $f:X\rightarrow Y$ is a continuous map of topological spaces and $\mathcal{F}$ is a sheaf (of sets for simplicity) on $X$. Denote the category of sheaves on $X$ by $\operatorname{Sh} (X)$. We can obtain a sheaf on $Y$ as follows; if $U\subseteq Y$ is open, define $f_{*} \mathcal{F} (U) = \mathcal{F} (f^{-1} (U))$. Clearly this is a functor $\bar{f}_{*} : \operatorname{Sh} (X) \rightarrow \operatorname{Sh} (Y)$ taking each sheaf to its direct image under $f$ and each $X$-sheaf morphism to a $Y$-sheaf morphism which commutes with the direct image.

Using this, we can actually produce a much more general functor into a huge category - the category of all categories of sheaves on (small) topological spaces; define $\operatorname{SH} (-) : \bf{Top_{small}}\rightarrow \bf{Sheaves}$ to be this functor, taking each small topological space $X$ to its category of sheaves $\operatorname{Sh} (X)$ and each continuous map $f:X\rightarrow Y$ to the direct image functor $\bar{f}_{*} : \operatorname{Sh} (X)\rightarrow \operatorname{Sh} (Y)$ defined above.

Have I interpreted this correctly? And if so is there any real/interesting usage (in algebraic geometry/category theory or other areas) of this functor $\operatorname{SH} (-)$ or this category $\bf{Sheaves}$ of all sheaves over topological spaces?

• A quick comment supplementing Pece's precise answer: Yes, the functoriality of $\mathrm{SH}(-)$ is used all over the place in algebraic geometry, in the form that $f_*(g_*(\mathcal{E})) \cong (f \circ g)_*(\mathcal{E})$ and in the form that $g^*(f^*(\mathcal{F})) \cong (f \circ g)^*(\mathcal{F})$. – Ingo Blechschmidt Sep 2 '18 at 11:07

Say you work with a universe $\mathbb U$, and you call small the elements of $\mathbb U$. Then, on one hand, the category of small topological spaces is a locally small category (meaning that the hom-sets are small). On the other hand, $\mathbf{Sheaves}$ is not locally small : indeed, for a small topological space $X$, the category $\operatorname{Sh}(X)$ isn't small (only locally small) ; so for small topological spaces $X$ and $Y$, the set of functors $\operatorname{Sh}(X) \to \operatorname{Sh}(Y)$ has no reason to be small.
So you end up with a jump of universe. Namely, there exists a universe $\mathbb V$ such that $\mathbb U \in \mathbb V$ making $\mathbf{Sheaves}$ a locally $\mathbb V$-small category. So your functor $\operatorname{SH}(-)$ is going from the locally $\mathbb U$-small category $\mathbf{Top_{small}}$ to the category the locally $\mathbb V$-small category $\mathbf{Sheaves}$. As any $\mathbb U$-small set is also $\mathbb V$-small, this is not a problem and $\operatorname{SH}(-)$ could be regarded as a functor between locally $\mathbb V$-small categories.
(Quick remark : one would usually think of $\operatorname{SH}(-)$ as a functor valued in the category $\mathsf{CAT}$ of all $\mathbb V$-small categories, or in the category of $\mathbb V$-small (elementary) toposes.)
• @XeSaad Yes, I was thinking of Grothendieck universes, but you can take any notion of universes and it will work more or less the same. The important part is to understand that your functor $\operatorname{SH}(-)$ can not live at the same level as your categories $\operatorname{Sh}(X)$ or $\mathbf{Top}$. If you want to read about Grothendieck universes, you can take a look at the very beginning of SGA4. – Pece Oct 23 '14 at 11:16