This is a cross-post from MSE.
$\require{AMScd}$The following proposition (1.13) is given in Tannakian Categories (loosely paraphrased with some change in notation).
Assume that $(F,c),(G,d):\mathcal{C} \rightrightarrows \mathcal{C}'$ are tensor functors. If $\mathcal{C}$ and $\mathcal{C}'$ are both rigid tensor categories, then every morphism of functors (my comment: natural transformation, I presume) $\lambda:F \Rightarrow G$ is an isomorphism.
The proof given in Tannakian Categories looks like this.
The morphism $\mu:G \to F$ making the diagrams\begin{CD} F(X^{\vee}) @>\lambda_{X^{\vee}}>> G(X^{\vee})\\ @VV\simeq V @VV\simeq V\\ F(X)^{\vee} @>{}^{t}(\mu_{X})>> G(X)^{\vee} \end{CD}commute for all $X \in \mathcal{C}$ is an inverse for $\lambda$.
Now, I feel quite lost in how to actually prove this. A footnote in the same article at the end of the page gives us that one can either use Yoneda lemma or the discussion here: Functors on rigid tensor categories.
The second link in turns gives us the following link: Morphism between tensor functors which has a diagram that seem quite messy. Any clarification on what to focus on to understand this proof would be appreciated.
Which proof is conceptually clearer; the one involving Yoneda lemma or working through the diagram in the last link (Morphism between tensor functors)?
Comment: Perhaps I should have waited longer before crossposting; my lack of patience in this regard got the better of me.