Is there a standard categorical counterpart of the construction, for some function f:A→B, elements a∊A, b∊B, of (f|a↦b) where (f|a↦b)(a) = b (f|a↦b)(x) = f(x) when x≠a ?
semantically this will often be impossible; for about as nice an example as i can think of consider the category of multigraphs (which is a presheaf category and so a topos). then you generally won’t be able to do this unless a is an isolated point.
