Knowlpedia

TikZ lab: adjunction naturality

A diagram stress test for adjunction bijections, units, counits, and LaTeX formulas.

For an F⊣GF\dashv G, the hom-set bijection

ΦC,D:D(FC,D)⟶C(C,GD)\Phi_{C,D}:\mathcal D(F C,D)\longrightarrow \mathcal C(C,G D)

is natural in both variables. If u:C′→Cu:C'\to C and v:D→D′v:D\to D', naturality is encoded by the square

ΦC′,D′(v∘f∘Fu)=Gv∘ΦC,D(f)∘u.\Phi_{C',D'}(v\circ f\circ F u)=Gv\circ \Phi_{C,D}(f)\circ u.
Rendered tikz-cd diagram

The unit and counit recover the bijection by

ΦC,D(f)=Gf∘ηC,ΦC,D−1(h)=εD∘Fh.\Phi_{C,D}(f)=Gf\circ\eta_C,\qquad \Phi_{C,D}^{-1}(h)=\varepsilon_D\circ Fh.
Rendered tikz-cd diagram
Rendered tikz-cd diagram

The following raw TikZ picture exercises curved arrows and a central formula inside a framed diagram.

Rendered tikz diagram