Comparion theorem between symmetric monoidal $\infty$-functor

Question

Let $T,T'$ be symmetric monoidal $\infty$-categories. And let $F_1,F_2:T\to T'$ be symmetric monoidal functors and let $t:F_1\Longrightarrow F_2$ be a symmetric monoidal natural transformation from $F_1$ to $F_2$. Antieau-Mathew-Nikolaus prove that if every object of $T$ is dualizable, then $t$ is an equivalence.

I have a question about this.

Suppose the unit $1_{T'}$ of $T'$ is the initial object. For any symmetric monoidal functor $F:T\to T'$, there is a symmetric monoidal natural transformation from $\hat{1}_{T'}\Longrightarrow F$, where $\hat{1}_{T'}$ sends every object to the unit $1_{T'}$. Thanks to Antieau-Mathew-Nikolaus, if every object of $T$ is dualizable, then $t$ is an equivalence, i.e. $F(x)\simeq 1_{T'}$ for any $x\in T$.

Why does this strange phenomenon occur? It should not happen in ordinary category theory.

Answer 1

If every object is dualizable and $1$ is initial, then your category is trivial. Indeed, for two arbitrary objects $x, y$, $$ \operatorname{Map}(x, y) \simeq \operatorname{Map}(1, x^\vee \otimes y), $$ and so there is a unique morphism between any two objects.