I feel that this question is interesting but has not received enough attention; possibly because it's in MSE. So, the present question is mainly a repost, in the hopes of getting a good answer. (If that's not allowed, I'll gladly delete the post.)

The aforementioned question asks which functors deserve the name "morphisms of Tannakian categories".

Let $F:\mathsf{C}\to\mathsf{D}$ be a functor between $k$-linear Tannakian categories. I've seen people say that $F$ is a morphism of Tannakian categories when:

1. (N. Saavedra Rivano, M. D'Addezio and H. Esnault) $F$ is $k$-linear, exact and (strongly) monoidal (compatible with the associativity and commutativity constraints and with the unit isomorphisms). (Such a functor is automatically faithful.)
2. Same as 1. but also supposing that if $Y$ is a subobject of $F(X)$, then there exists a subobject $Y'$ of $X$ such that $F(Y')\cong Y$.
3. (J. Lurie) $F$ is a continuous, tame, additive tensor functor. (Check Def. 5.9 in this paper for the precise definition.)

So, following the other question, I ask: which properties should define a morphism of Tannakian categories? And why?

References: N. Saavedra Rivano's thesis Catégories Tannakiennes, M. D'Addezio and H. Esnault's paper On the Universal Extensions in Tannakian Categories, J. Lurie's paper Tannaka Duality for Geometric Stacks.