What is "prestack"

n. (context category theory English) The category ''c'' is called a prestack over a category ''C'' with a Grothendieck topology if it is fibered over ''C'' and for any object ''U'' of ''C'' and objects ''x'', ''y'' of ''c'' with image ''U'', the functor from objects over ''U'' to sets taking ''F'':''V''→''U'' to Hom(''F''*''x'',''F''*''y'') is a sheaf.