Special cases of Lie II for groupoids using elementary techniques I asked a similar question on math.stackexchange but did not get any responses, so I thought I'd kick it up to mathoverflow.
In Crainic and Fernandes's "Integrability of Lie Brackets" (and the accompanying lecture notes), they use Lie II to prove the equivalence between $A$-paths (Lie algebroid morphisms $TI \to A$, where $TI$ is the tangent bundle over the unit interval) and $G$-paths (functors from the pair groupoid over $I$ into $G$). This is, of course, rock solid as a proof, but I find it a bit unsatisfying to have a fairly important step of a proof handled by applying a very powerful theorem to one of the simplest possible cases where it applies.
I first tried unwinding one of the original proofs of Lie II for groupoids from Moerdijk and Mrcun's book, where I assumed the source-simply-connected Lie groupoid in question is the pair groupoid over $I$. However, I once again found myself using powerful theorems about foliations while dealing with one of the simplest possible cases (the foliation of $I \times I$ by $I$). This feels like a result that should be amenable to elementary techniques and is nestled away in a masters thesis somewhere or a paper from the early days of Lie algebroids, but I can't seem to find a reference. I'm also sure that someone working in this field can probably pull the construction off the top of their head, but I'm having trouble finding the construction myself.
In summary:

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*Does anyone know where I can find an explicit/direct proof of the bijection between functors on the pair groupoid over the unit inverval $I$ into a Lie groupoid $G$ and Lie algebroid morphisms $TI \to Lie(G)$?

*It seems as though the construction in (1) should be derivable from something like "the pair groupoid over a vector space $V$ integrates the trivial lie algebroid $TV \cong V \times V \xrightarrow{\pi_0} V$". This would be the ideal construction to find.

Update 1: The natural next step is to consider the path integration method on a Lie group. There is an explicit proof of the bijection in Proposition 1.13.4 of Duistermaat and Kolk's Lie Groups.
 A: I suspect you've probably figured it out by now but here is an answer anyway:

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*Here is a sketch of the most low-tech way of accomplishing the task.

First, there is a bijection between groupoid homomorphisms $\Gamma \colon I \times I \to G$ and smooth functions $\gamma \colon I \to G$ such that $\gamma(0)$ is a unit and $\gamma$ is tangent to the source distribution $\ker d s$. This bijection is as follows:
$$ \Gamma(t_1, t_0) := \gamma(t_1) \gamma(t_0)^{-1} \qquad \gamma(t) := \Gamma(t,0) $$
Now if you are given a path $\gamma \colon [0,1] \to G$ which is tangent to the source fibers and such that $\gamma(0)$ is a unit, then one can construct a function $a \colon [0,1] \to A$ by:
$$ a(t) = d L_{\gamma(t)}^{-1} \gamma'(t) $$
where $L_g \colon s^{-1}(t(g)) \to s^{-1}(s(g))$ is the left translation map.
Perhaps the trickiest part is to show the correspondence $\gamma \to a$ ends up being a bijection. Surjectivity is a consequence of existence of solutions to ODEs and infectivity is a combination of the assumption that $G$ is source-simply-connected and uniqueness of solutions to ODEs.


*the result you are asking for is not proved but appears in Lectures on Integrability of Lie Brackets by Crainic and Fernandes as exercise 30. The exercise is not difficult:

The source map of the pair groupoid $M \times M$ over $M$ is just projection to the second component. The identity section is the diagonal map $\Delta \colon M \to M \times M$ and there is an obvious identification of $TM$ with $A := \ker d \pi_2 |_{\Delta(M)}$. The target map is projection to the first component which means that the anchor map $\rho \colon A \to TM$ is a vector bundle isomorphism. Since $\rho$ preserves the Lie bracket of both algebroids it is an algebroid isomorphism.
To show that the Lie algebroid of the fundamental groupoid of $M$ is $TM$ you can accomplish this by observing that $(pi_1 \times \pi_2) \colon \Pi_1(M) \to M \times M$ is a groupoid homomorphism and a local diffeomorphism.
