So we have topological spaces A,B and sheaves F,G on A,B of vector spaces over some fixed field k and want to construct a sheaf AotimeskB on the product space AtimesB. You can write it down explicitly:
Let WsubseteqAtimesB be open. Then (FotimeskG)(W) consists of those elements sinprod(a,b)inWFaotimeskGb, such that for all (a,b)inW there are open sets ainUsubseteqA,binVsubseteqB and tinF(U)otimeskG(V) such that UtimesVsubseteqW and for all (c,d)inUtimesV, we have tc,d=sc,d. Here tmapstotc,d denotes the canonical map F(U)otimeskG(V)toFcotimeskGd.
Note that this obviously(!) yields a sheaf on AtimesB. On stalks, there is a canonical map (FotimeskG)a,btoFaotimeskGb; a calculation shows that it is bijective. Remark that this agrees with the definition given by Strom Borman (the same universal property holds). But here you have a description of the sections of FotimeskG. In particular, you see that if F and G are the sheaves of mathbbK-valued continuous functions on A resp. B, then FotimesmathbbKG is a rather small subsheaf of the continuous functions on AtimesB.
The whole things makes more sense, when we take A,B to be two S-schemes (or more generally, locally ringed spaces). Then we have the fibred product AtimesSB which can be constructed as above (I've written this up here (in german)). Here, the tensor product is the "right" sheaf.
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