Perhaps Lie theory is not the correct term, but I’m thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie algebroid of $G$ by taking the pullback of the cospan

$$ M xrightarrow{(e,0)} G times TM xleftarrow{(p, Ts)} TG$$

Now, if $G$ were just a reflexive, involutive graph rather than a groupoid, we would still get a vector bundle over $M$ from this pullback, with an anchor $varrho := A hookrightarrow TG xrightarrow{Tt} TM$.

Has anyone written down the analogue of “Lie integration” for this setting? If I move to synthetic differential geometry there’s a way through using enriched sketches, but I’d like to see something a little more down to earth.