I can't seem to wrap my mind around what is going on when I try to visualize $Fct(Op_X, Set)$, as one example. Now I know that a functor is a morphism between categories hence we have a morphism between the category of open sets of $X$ and the category of sets.
What does this mean, intuitively speaking what is really going on here and can anyone help me (and others that are finding some of this category theory quite abstract), visualize this?
Edit: The image below is my ideas of what is going on. Note, ()'s are open sets, |'s are for closed sets. I don't know if denoting the functor as $F^{op}$, but what I imagine is that since U is an "inclusion" in V as $U \subset V$, that there are morphisims $r$ which map elements in $U$ to those same elements in $V$ - so as to get an inclusion i.e. it is these morphisms that create the inclusion. The like idea is the same with $F(V) \rightarrow F(U)$ as although $F(U) \subset F(V)$, we are dealing with the opposite category open sets of $X$. Please let me know if anything in the diagram is wrong.
Thanks in advance,
Brian
From the comments, I assume you're working on presheaves and so you're interesting in functors ${\mathcal O_X}^\mathrm{op} \to \mathbf{Set}$ rather than ${\mathcal O_X} \to \mathbf{Set}$ ($\mathcal O_X$ being the category of open sets of $X$).
First, just recall that $\mathcal O_X$ is the category with objects the open sets $U$ of the topological space $X$, and with morphisms the inclusions of open sets : it means that considering $U,V$ open sets of $X$, $$ \mathrm{Hom}(U,V) = \left\{ \begin{aligned} \{ \ast \} \quad &\text{if $U \subseteq V$} \\ \emptyset \quad &\text{otherwise.} \end{aligned} \right. $$
So a presheaf $F \colon {\mathcal O_X}^\mathrm{op} \to \mathbf{Set}$ is the data of a set $F(U)$ forevery open set $U$ of $X$, and of a set function $F(V) \to F(U)$ for every inclusion $U \subseteq V$ of open sets in $X$. This data is required to satisfy : for open sets $U \subseteq V \subseteq W$ of $X$, the composite set function $F(W) \to F(V) \to F(U)$ is the function $F(W) \to F(U)$, and the function $F(U) \to F(U)$ coming from the trivial inclusion $U \subseteq U$ need to be the identity function.
With the hands, a presheaf is a set that evolves continuously along the topological space $X$. Especially, that set is compatible with zooming in the space $X$.
Mostly, nice visual examples of presheaves are sheaves (or at least separated presheaves). The most popular example is certainly the following : take a topological space $X$ (a topological variety is nice for visualisation), then the functor $U \mapsto \mathcal C(U,\mathbb R)$ of the continuous maps from $X$ to $\mathbb R$ is a presheaf (and even a sheaf) ; the set function $\mathcal C(V, \mathbb R) \to \mathcal C(U, \mathbb R)$ is just the restriction of the continuous maps defined on $V$ to $U \subseteq V$.
An other example : take again a topological variety $X$, the functor $U \mapsto \mathcal C_b(X, \mathbb R)$ of bounded continuous maps from $X$ to $\mathbb R$ is again a presheaf (actually a separated one).