Answer to Baez Charles Ehresmann described examples of what would later become a differentiable category and a differentiable groupoid in the early fifties, in the course of his development of fibred bundles and of Differential Geometry based on his notion of "jets", But it is only in his paper "Categories topologiques et categories differentiables" (CBRM 1959) that he gave a formal definition of a differentiable category, of a differentiable groupoid, and of the differentiable action of a differentiable groupoid looked at as a generalization of fibred bundle. Notice that nobody had yet thought of defining internal categories at this time... In this paper, he defines it as a (small) category whose set of morphisms C is equipped with a differentiable structure such that: (i) the domain and codomain maps a and b (he called them "source" and "target") are differentiable and of constant rank; (ii) this ensures that the set of composable pairs defines a sub-manifold of the product manifold CxC and the composition law must be differentiable from this to C. This same definition (still with a and b of constant rank) is recalled as an example in his important paper "Categories structurees" (Ann. Ec. Norm. Sup. 80, 1963), where he introduced the general notion of a "structured category", in modern terms an internal category in a concrete category (still at this date, internal categories had not been considered...), with e.g. the example of a double category. And it is extensively used in his paper "Prolongements d'une categorie differentiable" (Cahiers Top. et Geom. Diff. VI, 1964) and some subsequent papers. Now in1966, in the paper."Introduction to the theory of structured categories" (Tech. Report 10, Univ. of Kansas at Lawrence, 1966), he introduced the "sketch of categories" and, as a model of this sketch, the (now usual) notion of an internal category in any category under the name "categorie structuree generalisee". Thinking about this general notion, he naturally came to the idea that what was important in the case of differentiable categories was not the fact that a and b are submersions but only that there exists a pullback. And it is the definition that he gave in his later courses. He also tackled this problem in another way, by trying to extend the category of differentiable manifolds by adding pullbacks, leading to a series of papers on dfferent general completion theorems, in particular in the paper "Prolongement universel de foncteurs par adjonction de limites" (Dissertation. Math.LXIV, 1969), where the differentiable case is explicitly refered to. All the cited papers are reprinted in "Charles Ehresmann Oeuvres completes et commentees", Parts I to IV (Amiens, 1980-83), where many historical indications are given in the long comments I have written. Cf; also my paper "From fibre bundles to categories" (Cahiers XXV-1, 1984). Sincerely Andree C. Ehresmann