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There are results in category theory that imply that some morphism is invertible when a priori one might not have expected it. For instance,

  1. Given a monoidal natural transformation $\tau$ between strong monoidal functors (or more generally, we can work with Frobenius monoidal functors ), the component of $\tau$ at any dualizable object is invertible.
  2. Given a lax monoidal functor $F$ with a lax monoidal right adjoint, $F$ is in fact strong monoidal (and other instances of doctrinal adjunction).
  3. In some specific situations a given canonical morphism is invertible provided any non-canonical isomorphism exists. For instance, if a category has a zero object, binary products and binary coproducts, then the canonical natural transformation $(-)\coprod (-)\to (-)\times (-)$ is an isomorphism as soon there exists any natural isomorphism between these functors.

Do you know other surprising invertibility results that would fit the list? I'm primarily interested in categorical situations, but examples from other fields are welcome as well.

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    $\begingroup$ Something similar happens for the unit and counit of an adjunction, iirc: if $FG\cong 1, GF\cong 1$ by some isomorphisms, then the right ones, unit and counit, are also invertible; follows from naturality. $\endgroup$
    – fosco
    Nov 23, 2021 at 7:24
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    $\begingroup$ @fosco That's an answer, isn't it? $\endgroup$ Nov 29, 2021 at 20:57

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For a monoidal category $(\mathbb C,\otimes,I)$, if you want all objects being (functorially) weakly invertible wrt the monoidal product, then your category $\mathbb C$ is in fact a groupoid, i.e. all arrows are strictly invertible wrt composition: invertibility of arrows follows for free!

Notice that the required functoriality of the inversion map $$ (\ )^*\colon \mathbb C \to \mathbb C $$ is a natural assumption, since it makes the monoidal category (which is already a weak monoid object in $Cat$) a weak group object in $Cat$.

This is why categorified groups, aka weak $2$-groups, are monoidal groupoids.

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