Integrability of Lie algebroids In the article https://arxiv.org/pdf/math/0611259.pdf, it is defined the integrability of a Lie algebroid as follows: a Lie algebroid $A$ is integrable iff it is isomorphic to the Lie algebroid of a Lie groupoid $\mathcal{G}$. I have two questions concerning this definition:
1) The Lie groupoid $\mathcal{G}$ is required to be over the same base as $A$?
2) What is exactly an isomorphism of Lie algebroids? It is strange, because they define this notion just before defining what a morphism of algebroids is. Is it because the isomorphism is intended to be over the same base, as I'm asking in 1), and then the notion of compatibility with the anchors and the brackets is trivial and, hence, such an isomorphism is just an isomorphism of vector bundles over the same base with these compatibilities?
Thanks a lot!
 A: 1) Yes: the Lie algebroid associated to a Lie groupoid $G\rightrightarrows M$ is a vector bundle over $M$. If we'd like a Lie groupoid to be an integration of a given Lie algebroid, then its associated Lie algebroid should be the one we started with.
2) The definition of what it means to be a (general) morphism of Lie algebroids is not obvious because a map of vector bundles does not, in general, induce a map on sections. If, however, we have a map of vector bundles whose base map is a diffeomorphism, then there is an induced map on sections - so indeed, checking that an isomorphism of vector bundles is an isomorphism of Lie algebroids is easier than the general case.
To be a bit more precise, suppose we have two Lie algebroids $A\to M$ and $B\to N$, and $\varphi\,\colon A\to B$ is a map of vector bundles such that the base map $f\,\colon M\to N$ is a diffeomorphism. Then, since $f$ is invertible, there is a pushforward map $\varphi_*\,\colon\Gamma(M,A)\to\Gamma(N,B)$. Then it is true that $\varphi$ is a morphism of Lie algebroids if it is compatible with the anchors (which means $\rho_B\circ \varphi = df\circ \rho_A$) and $\varphi_*$ is compatible with the Lie brackets. In particular, $\varphi$ is an isomorphism as long as it's an isomorphism of vector bundles and these two conditions are satisfied.
