Let {\mathcal{C}} be a monoidal category—I don’t actually want to define what that means, so I refer you to  John Armstrong’s post–with {\otimes} as the monoidal operation. Suppose {1} is a unital object.

The following is well-known:

Theorem 1 {End(1)} is a commutative monoid.

 

Oftentimes, one has an additive structure on {\mathcal{C}} as well, and one actually wants {End(1)} to be a field. The result is interesting, because it strikes a parallel with the following:

Proposition 2 The endomorphisms of the identity functor in a category {\mathcal{C}} form a commutative monoid.

 

The proof is different though.   In some places, it’s not even properly mentioned; in others, it’s always seemed extremely non-intuitive.

I learned of a neat proof of the first theorem in the first chapter of a book by Saavedra on Tannakian categories. It is as follows. By definition, {1 \simeq 1 \otimes 1}, so it is enough to prove {End(1 \otimes 1)} commutative. Let {f, g \in End(1 \otimes 1)}. Since the functors {- \rightarrow - \otimes 1} and {- \rightarrow 1 \otimes -} are equivalences of categories, and in particular fully faithful, we can write {f = u \otimes \mathrm{id}_1, g = \mathrm{id}_1 \otimes v} for appropriate {u,v \in End(1)}. But then

\displaystyle f \circ g = u \otimes v = g \circ f ,

which proves commutativity.

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