If I've understood it right, this is exactly the concept of a "multi-terminal object" (that is, a multi-limit for the empty diagram). The name is due to me (in a paper called "A syntactic approach to Diers's localizable categories" in SLNM 753 (1979)), but the concept is due to Yves Diers: see his "Familles universelles de morphismes", Ann. Soc. Sci. Bruxelles 93 (1979). Peter Johnstone On Fri, 4 Dec 2009, Ellis D. Cooper wrote:
Dear categorists,
Let C be a category with a distinguished sub-category E and a distinguished family S of morphisms such that for every object x of C there is a unique morphism f_x: x ---> e_x with e_x an object of E so that the following conditions are satisfied: (1) if x is in E then f_x = 1_x (the identity morphism of x), (2) if s: x ---> y is in S then e_y = e_x and f_x = f_y s.
Hasn't this simple situation been named and incorporated in some publication on category theory? A reference would be most appreciated.
Ellis D. Cooper
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