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A modern point of view on such lemmas in abelian categories makes systematic use of the salamander lemma. See for example sbseminar.wordpress.com/2007/11/13/… (by MO founding member Anton Geraschenko).
No, they're not different. But that doesn't undermine the point being made. (Also, I imagine that what FDHilb is taken to mean might vary with author. For example, for some contexts it seems sensible to consider not all linear maps, but those whose norm doesn't exceed $1$. See this old nForum thread, comment #17 particularly about this point: nforum.ncatlab.org/discussion/3158/…
A small note: if $X^A$ exists for $X$ the Sierpinski space, then $A$ is exponentiable. This doesn't yet address the situation with $X^A$ existing for all compact Hausdorff $X$, but for what interest it has I'm mentioning it.
Qiaochu is saying that as far as the chosen morphisms (general linear maps) are concerned, they do not reflect or detect anything about Hilbert space structure, and so categorically speaking that structure becomes irrelevant with that choice. For all the morphisms are concerned, you might as well be talking simply about vector spaces.
