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On nLab, there are some string diagrams for the triangle identities for the unit and counit of an adjoint pair $(L,R,\eta,\epsilon)$.

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and with minimal notation,

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I'm confused about how to read these, for in the second and fourth pictures, following the arrow seems to say that $R\epsilon\circ \eta R=1_R$, which is one of the usual triangle identities in equation form, but following the arrow in the third picture, the labels in the first picture seem to indicate an identity where the counit $\epsilon$ is applied before the unit $\eta$ to get $1_L$ somehow, but both triangle identities I know always have the unit coming first. Am I reading the diagrams incorrectly?

Page with the pictures is: https://ncatlab.org/nlab/show/triangle+identities#AsStringDiagrams

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  • $\begingroup$ arxiv.org/abs/1401.7220 discusses string diagrams for 1-categories in detail. In that article, diagrams are read bottom-up. With a bottom-up reading, this should be reasonable. Functor composition is in diagrammatic order, i.e. $L\circ R$ would have $R$ on the left and $L$ on the right. $\endgroup$ – Derek Elkins Nov 1 '18 at 2:15
  • $\begingroup$ @DerekElkins Thanks, I wasn't sure since I tend to read the diagrams bottom up, and right to left. Thanks also for the reference. $\endgroup$ – Hailie Mathieson Nov 1 '18 at 6:41
  • $\begingroup$ There are also quite a few references (including the paper mentioned by Derek) at this page : ncatlab.org/nlab/show/string+diagram#references. $\endgroup$ – Arnaud D. Nov 1 '18 at 12:15
  • $\begingroup$ The arrow direction is just to denote whether you're in $R$ or $L,$ not at all to describe composition. $\endgroup$ – Kevin Carlson Nov 1 '18 at 18:13

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