Midoriya has less of an eye for the history of a place, and more concern with safety, structural integrity, and watching out for black mold. Nonetheless, he's habitually respectful of things that seem set apart. He moves things carefully, and takes care not to damage anything. He's not religious, but it's baked into where he comes from (Japan, roughly modern-day) and his innate politeness that you just don't mess around in another god's house or with various shrines. He's here to keep people safe, spruce up the place, and pay respects wherever they're due so as to avoid being smote. The gods are multitudinous and innumerable.

For a moment, he's afraid they trespassed, and this is a minor god or guardian telling them to step off. Then Palamedes speaks like a scientist in the midst of all this. Midoriya recalls his explanation of life and death energies, and, dimly, recommended videos he saw online. Palamedes can see more than he can, and ghost hunting videos are pretty popular.

It is then Midoriya remembers he probably cannot punch a ghost.

"Definitely trying to attack, we need to hide, come on--"

The ghost isn't raising an arm, but it (they?) moves at a fast stride. Midoriya throws an arm around Palamedes's middle and tries to encourage him to run of his own accord. It's either that or picking him up.