How do you define the tangent Lie algebroid to a Lie groupoid?
In this online note Lie Algebroids, Lie Groupoids and Poisson Geometry by Sébastien Racanière, it states that if $t\colon G_1\to G_0$ is the target map from arrows to objects of the Lie groupoid, then since it is a submersion, its kernel is a vector bundle $T^tG_1\to G_1$. But the target map $t$ is just a smooth submersion, and its domain. Does he mean kernel of the differential of the target map $dt\colon TG_1\to TG_0$?
Also, once we have a definition of the tangent Lie algebroid, how do we induce a homomorphisms of Lie algebroids, starting from a homomorphism of Lie groupoids?