About the sheafification

Question

Look at the following definition of the shefification (the source is the Stacks Project):

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I don't understand what is the projection $\prod_{u\in U}\mathcal F_u\longrightarrow\prod_{v\in V}\mathcal F_v$. In particular, if $u\in U\setminus V$ what is the image of $s_u$ under this map?

Thanks in advance.

Answer 1

The actual question turns out to not be particular to sheaves, so I will adjust my notation accordingly. The elements of a product $\prod_{a\in\mathcal A}X_a$ are tuples (well, choice functions) $(x_a)_{a\in\mathcal A}$. If $\mathcal B\subset\mathcal A$, then we can assign $$(x_a)_{a\in\mathcal A}\mapsto (x_b)_{b\in\mathcal B}.$$ This map $\prod_{a\in\mathcal A}X_a\to\prod_{b\in\mathcal B}X_b$ is what the author was referring to.

Also, note that you can use the universal property of products in categories to determine the corresponding projections in non-concrete categories.