Yes, I agree with that. In some situations, one may add "coherence conditions" which can serve to pin down the "right" natural isomorphism, as for example the associativity isomorphism $\alpha: (X \otimes Y) \otimes Z \to X \otimes (Y \otimes Z)$, but that's of course something extra.

It most certainly was a fair objection, and you mischaracterized what the objection was. It was that following the method of the OP, one defines two different functors from two different collections of isomorphisms. It wasn't that you couldn't get two different natural isomorphisms between two functors given in advance. In fact, if $\phi, \psi: F \to G$ are two different natural isomorphisms between $F, G$ given in advance, then one gets the same $G$ defined by conjugating $F$ by $\phi$ or by $\psi$ -- quite unlike the situation I was describing.

"Why not? I've followed exactly the same recipe for both of them, using the notion of "natural transformation" as I was supposed to." Cf. my answer. Who said that particular recipe was the one you were supposed to follow? In the end, do any two cooks write down the same recipe?

I am rolling back to the previous revision, which reflects what was said in the last comment by Hussain Rashed. If Hussain disagrees with that revision, then he may clarify.

Comments are not for extended discussion; this conversation has been moved to chat.

Roughly speaking, my understanding is that if you have two $n$-dimensional structures (meaning that the top dimension of non-identity cells in each is $n$), then under an equivalence, a pair of mutually inverse $n$-cells in a local hom-category in one of the structures will get mapped to a pair of mutually inverse $n$-cells in a local hom-category of the other. The effects of this propagate down to lower-dimensional cells that participate in higher equivalences.

I don't see a problem considering a construction like $yB = \text{LaxBiCat}(B, Cat)$, but the exact features would be something to work out. I'll bet Nick Gurski could tell you more. But I do have a hard time believing that a notion of equivalence by which lax constraints are rendered 'pseudo' would be deserving of the name 'equivalence'.

In the argot I know, a weak $n$-category refers to the sequence set, category, bicategory, tricategory, etc., and so a weak 2-category would be a bicategory (although it may be rare these days to hear it referred to as such). There may be lax bicategories, where the one-object case is sometimes called a skew monoidal category. But I thought the Yoneda lemma applied to bicategories (?). Do you have a literature reference for your last paragraph?

jorge, this is not a proper format for an MO answer. First, references to the literature need to be precise in order to be helpful; if you can point to specific results or to page numbers in the article, that's even better. But mostly what question-posters want to get is a to-the-point, informative, and informed answer from someone with expertise, not some vague hand-waving. Please see Mark's answer above for an idea of a proper response.

@AsafKaragila In fact it didn't occur to me to read it any other way.

@TimCampion Tim, yes, I assume we are talking about the large poset that is the posetal collapse of the preorder defined by $X \leq Y$ iff there exists an injection from $X$ to $Y$ (by Cantor-Schroeder-Bernstein, $X \leq Y$ and $Y \leq X$ iff $X$ and $Y$ are isomorphic). The question is certainly sensible under this assumption.

I really don't get the sense of the question at all. It is well known that a topological space can be equivalently defined to be a set $X$ equipped with a closure operation $P(X) \to P(X)$ that preserves finite unions. This, or its interior coclosure counterpart, play a role in various places such as intuitionistic logic where topologies provide fundamental examples of Heyting algebras. But I don't see how the complementation operator deserves to be called 'topological'. I think we need a Russophone to explain the text in question and its significance.

Marion, to my eyes, there was no sarcasm or mockery. In fact, I see it as expressing sympathy and commiseration.

Welcome to MO, Professor Holmes!