If X is a topological space and (U$_i$)$_i$$_\in$$_I$ is a family of open subsets of X, write U = $\cup$$_i$$_\in$$_I$ U$_i$ and U$_i$$_j$ = U$_i$ $\cap$ U$_j$ for i, j $\in$ I.
(i) if f, g: U $\to$ R are continuous functions such that f|U$_i$ = g|U$_i$, then f = g
(ii) if (f$_i$: U$_i$)$_i$$_\in$$_I$ is a family of continuous real-valued functions such that f$_i$|U$_i$$_j$ = f$_j$|U$_i$$_j$ for all i, j, then there exist a unique continuous f: U $\to$ R with f|U$_i$ = f$_i$ for all i.
For every oepn V $\subseteq$ X, define $\varGamma$(V) = {continuous f: V$\to$ R} and $\varGamma$(W) $\to$$\varGamma$(V) to be the restriction for V a subset of W. Then properties (i)(ii) say that $\varGamma$(U) is the equalizer of the family of maps $\varGamma$(U$_i$) $\to$$\varGamma$(V$_i$).
I am confused by the last sentence. How (i)(ii) give that conclusion? What are the two maps $\varGamma$(U$_i$) $\to$$\varGamma$(V$_i$) which consist of part of the equalizer? And I take it the map from $\varGamma$(U) $\to$$\varGamma$(U$_i$) is the restriction?
I've also tried thinking about it as presheaf functor, still not quite clear. Thank you SO MUCH!
In the category of vector spaces we have the following diagram:
$$\prod_{i\in I}\Gamma(U_i)\begin{array}{c}\stackrel \alpha\longrightarrow\\ \stackrel \beta \longrightarrow \end{array}\prod_{(i,j)\in I\times I} \Gamma(U_{ij}),$$ where \begin{eqnarray*}\alpha(\{f_i\}_{i\in I})&=& \{f_i|U_{ij}\}_{i,j},\\ \beta(\{f_i\}_{i\in I})&=& \{f_j|U_{ij}\}_{i,j}. \end{eqnarray*}
From (i), (ii) we know that the equaliser of this diagram is $\Gamma(U)$:
$$\Gamma(U)\stackrel\iota\longrightarrow\prod_{i\in I}\Gamma(U_i)\begin{array}{c}\stackrel \alpha\longrightarrow\\ \stackrel \beta \longrightarrow \end{array}\prod_{(i,j)\in I\times I} \Gamma(U_{ij})$$ Here $\iota(f)=\{f|U_i\}_i$.