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Hi learning Lean I came cross the following property of category equivalence:

Definition An equivalence of categories cosnists of $(F, G, \eta, \epsilon)$ where $F: C \rightarrow D$ and $G: D\rightarrow C$ are functors and $\eta: 1_C \cong GF$, $\epsilon: FG\cong 1_D$ are natural isomorphisms.

and

Theorem if $F \Rightarrow FGF \Rightarrow F$ is identity, so is $G \Rightarrow GFG \Rightarrow G$.

The purpose of this theorem is for showing that the above definition can be equipped with the two triangle equalities in the above theorem, as said in the document of Lean's Mathlib4 Equivalence of categories.

I proved it manually but I don't get the proof linked to http://globular.science/1905.001, which shown as a diagram bellow, found in the source code in here. What does the meaning of it? How to translate it to a plain English proof? Thank you for any helps, ideas and references for it!

enter image description here

onRiv
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    That's what's called a "string diagram". I don't understand them either, but you can find an article that purports to explain them on nLab https://ncatlab.org/nlab/show/string+diagram – hasManyStupidQuestions Mar 31 '24 at 14:33
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    The nLab article itself is unhelpful, but maybe one of the references will be helpful. That having been said, I have tried several times before to find a good introductory reference to string diagrams and have yet to find one that is useful to me. The key idea seems to be that string diagrams can represent "monoidal categories" (and any "Cartesian product", i.e. "product" defined by the universal property of the product forms a monoidal category structure). – hasManyStupidQuestions Mar 31 '24 at 14:43
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    "It was proved by Joyal and Street that planar box-and-wire diagrams can unambiguously represent morphisms in a monoidal category. They showed furthermore than this representation is sound and complete with respect to the algebraic definition of a monoidal category." quote from p.21 of https://arxiv.org/pdf/1203.0202.pdf , the reference mentioned being: A. Joyal and R. Street. The geometry of tensor calculus I. Advances in Mathematics, 88:55–113, 1991 This reference https://arxiv.org/pdf/2305.08768.pdf seems like it might be helpful although possibly too detailed. – hasManyStupidQuestions Mar 31 '24 at 14:46
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    @hasManyStupidQuestions Thank you very much for your helps! I find some thread seems related to this now from https://math.stackexchange.com/questions/1789749/f-is-an-equivalence-of-categories-implies-that-f-is-fully-faithful-and-essen, which links to https://ncatlab.org/nlab/show/adjoint+equivalence I am trying to check f it's relavant or not.. – onRiv Mar 31 '24 at 14:49

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