  Lie algebroids are by no means natural as an infinitesimal counterpart of
groupoids. In this paper we propose a functorial construction called Nishimura
algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids,
intended for differential geometry, are of the same vein as Lawvere's
functorial notion of algebraic theory and Ehresmann's functorial notion of
theory called sketches. We study totally intransitive Nishimura algebroids in
detail. Finally we show that Nishimura algebroids naturally give rise to Lie
algebroids.
