  Ehresmann's introduction of differentiable groupoids in the 1950s may be seen
as a starting point for two diverging lines of research, many-object Lie theory
(the study of Lie algebroids and Lie groupoids) and sketch theory. This thesis
uses tangent categories to build a bridge between these two lines of research,
providing a structural account of Lie algebroids and the Lie functor.
  To accomplish this, we develop the theory of involution algebroids, which are
a tangent-categorical sketch of Lie algebroids. We show that the category of
Lie algebroids is precisely the category of involution algebroids in smooth
manifolds, and that the category of Weil algebras is precisely the classifying
category of an involution algebroid. This exhibits the category of Lie
algebroids as a tangent-categorical functor category, and the Lie functor via
precomposition with a functor $\partial: \mathsf{Weil}_1 \to
\mathcal{T}_{\mathsf{Gpd}},$ bringing Lie algebroids and the Lie functor into
the realm of functorial semantics.
