what does $s,t \in F(U)$ represent here?, Sheaf theory

Question

Wikipedia (https://en.wikipedia.org/wiki/Sheaf_(mathematics):

A sheaf is a presheaf with values in the category of sets that satisfies the following two axioms:

(Locality) If $\{U_i\}$ is an open covering of an open set $U$, and if $s, t ∈ F(U)$ are such that $s|_{U_i }= t|_{U_i}$ for each set $U_i$ of the covering, then $s = t$,

There is a 2nd condition but I don't write it here.

I want to understand what does $s,t \in F(U)$ represent here?

Are they object in category or simply element or functor?

Since $F(U)$ is an object, $t,s$ are element perhaps.

If $t,s \in F(U)$ are elements then how can we restrict liek $s|_{U_i}$ ?

Can you please explain what is $t,s \in F(U)$ and how restrictions are defined?

Answer 1

It is explained just a few lines above the one you quoted. A sheaf is first of all a presheaf, which includes

• For each open set $U$ of $X$, there corresponds an object $F(U)$ in $\mathbf C$.
• For each inclusion of open sets $V \subseteq U$, there corresponds a morphism $\operatorname{res}_{V,U}\colon F(U)\rightarrow F(V)$ in the category $\mathbf C$.

The morphisms $\operatorname{res}_{V,U}$ are called restriction morphisms. If $s \in F(U)$, then its restriction $\operatorname{res}_{V,U}(s)$ is often denoted $s|_V$ by analogy with restriction of functions.