I heard several times (for instance in these general lectures) that Grothendieck did functional analysis before he started doing algebraic geometry and category theory. It is said that at the time he did functional analysis, he didn't yet know what a category is, but his thinking was already very category-theoretic. For instance, I think McLarty roughly said in the linked lecture (although I don't recall the time stamp) that Grothendieck implicitly came up himself with the notion of a universal property and characterized some notions of tensor product in functional analysis via universal properties.
Question: Which theorems concretely did Grothendieck prove in functional analysis which are "category-theoretic" in spirit? How do these statements connect to more traditional functional analysis? (What I mean by the second question is for example: when he proved that some tensor products have a specific universal property, in which way did he apply that to solve concrete problems in functional analysis?)
I know that this question is not very concrete and rather historical (I apologize for that -- added the soft-question tag), but it's something I was wondering about when listening to these lectures, and maybe someone knows something interesting and more concrete to say.