Let $X,Y$ be two analytic spaces. Does the product $X\times Y$ exist? (in the category of analytic spaces and the category of locally ringed spaces)
I try to mimic the proof of schemes. We can do it locally, so can assume both affine. But for schemes we can do it because affine schemes and rings are equivalent categories so we can construct the product via tensor product of rings. So how to do this for analytic spaces?
Any hint or reference would be helpful.