Mr Byrne already answered, but that was my thought. If near enough the Batcave, he'd examine it; if not, and the Batmobile lab wasn't enough, he'd have to chalk it up as a loss.

One thing I loved about the "shared earth" aspect of early DC/Marvel crossovers was the comments made by the heroes, acknowledging each other's existences (in the second Superman/Spider-Man team-up, Wonder Woman mentions that she had been planning to hunt down the web-slinger).

Well, in the Batman/Hulk crossover, Batman tells someone (Commissioner Gordon, I think) that he had kept a file on the Hulk in the Batcave for quite some time. I imagine, if use the "shared earth" as a premise, Batman would keep files on all the heavy-hitters, including Spider-Man.

It's sort of why I dislike the "separate universes" premise that the modern crossovers did. Superman casually mentioning that he'd heard whispers about Galactus from heroes in another universe is not as fun as Batman telling Gordon about the Hulk's file in the Batcave.

When Spider-Man appeared in a TRANSFORMERS comic, it felt completely right. But years later, I did read comments from some, asking what earth it took place on. *Sighs*

The early DC/Marvel crossovers enhanced my enjoyment of superhero comics. I'd always wanted to believe they were on the same earth, but SUPERMAN VS THE AMAZING SPIDER-MAN confirmed it. I was pleased you went down that road with your DC/Marvel crossovers, plus the 1990s Hulk/Superman tale went down that same road.

I also think it's more accessible, e.g. Clark Kent and Peter Parker crossing paths due to journalistic responsibilities works better for me than bogged-down multiverse explanations involving cosmic entities. 

But my question, likely for another thread, is - how does Batman do with another "winner"? Some hero who is also knowledgeable and just about always ready to fight?

So of course... it's Batman vs Captain America again. Think I'll pop that up now...,