One doesn't need all the tau-category stuff just for finding an equivalent category with inclusion maps. There's a first-order construction but first an easier way to see how to do it. Given a category *A* define *B* to be the full subcategory of representable contravariant set-valued functors and all subfunctors thereof that are isomorphic to representable functors. The Yoneda embedding of *A* into *B* is clearly an equivalence functor. By saying that T is a subfunctor of (-,A) I mean that TX is a subset of (X,A) for each X and that the inclusions form a natural transformation, which transformations are to be, of course, the chosen notion of inclusion map in *B*. Done.. If one objects that this works only for small *A* use this first- order construction: take the objects of *B* to be the monics of *A*. The forgetful functor from *B* to *A* carries [A >-> B] in *B* to A in *A*. Take *B* to be the "inflation" that results from this onto forgetful functor, that is, the maps from [A'>-> B'] to [A >-> B] are named by maps from A' to A. Such a map is said to be an inclusion map iff A' >-> B' = A' -> A >-> B (in particular, A' -> A is monic and B' = B). Given [A'>-> B'], [A>-> B] and an arbitrary monic A' >-> A note that the identity map on A' names an isomorphism from [A' >-> B'] to [A'>-> A >-> B] such that the given monic [A' >-> B'] -> [A >-> B] is equal to [A' >-> B'] -> [A' >-> A >-> B] -> [A >-> B]. Define *A* => *B* to be the functor that sends each object in *A* to the object in *B* named by its identity map and define *B* => *A* to be the obvious forgetful functor. Then *A* => *B* => *A* is the identity functor on *A* and there's a natural equivalence from *B* => *A* => *B* to the identity functor on *B*. We don't need the axiom of choice to construct that natural equivalence; take each of its component isomorphisms as the one map that fits and that's named by an identity map. (The first would require the axiom of choice. The second construction is isomorphic to the category whose objects are ordered pairs, where the first coordinate is a subfunctor of a representable and the second is a specified isomorphism to a representable.) Mike asks how to pick what I'll call "inclusion maps" that is a choice of monics so that each subobject is named by a unique inclusion map and such that the inclusion maps are closed under composition. I don't know what Grothendieck had in mind, but it's an easy application of the stuff about tau-categories in Cats & Alligators copied from my 1975 "pamphlet." The setting there is cats with finite limits but of course you could always work in the full subcategory of representable functors in the category of set-valued contraviariant functors. A tau-structure gives you not only (transitive) canonical subobjects but associative canonical products and all sorts of goodies. But alas it delivers an equivalent category, not necessarily the category you started with. I don't know how to do it in an arbitrary category. But neither does anyone else, not even Grothendieck: Consider the subcategory of the category of sets with three objects, A,B,C each of which is a two-element set. There well be a total of ten maps, three of which, of course, are identity maps. The other seven maps will be one-to-one and onto but only one of them an isomorphism (as defined, of course, in the subcategory). The one non-trivial isomorphism will be on A (necessarily, the twist map). The hom-sets (B,B) and (C,C) each have a single element (necessarily their identity maps). The hom-sets (B,A), (C,A,) and (C,B) are empty. The hom-sets (A,B), (A,C) and (B,C) each have two maps, to wit, the two possible one-to-one onto maps. Note that all maps are monic. A has no proper subobjects. B has only one (the two monics from A to B name the same subject of B). C has three proper subobjects one of which is named by each of the two maps from A to C. Each of the other two subobjects of C are named by one of the two maps from B to C (since B has no non-trivial automorphism they necessarily name different subobjects). Hence, of the four proper subobjects in this category, two of them have just one name, the other two each have two names. No matter how one chooses a choice of "inclusion map" in those two equivalence classes, that is no matter how one chooses a map A -> B and a map A -> C, when you compose the chosen A -> B with each of the two maps from B to C you will get both names for the single proper subobject named by each of the two maps from A to C. They can't both be the chosen map in (A,C). [For admin and other information see: http://www.mta.ca/~cat-dist/ ]