The triangle identities or zigzag identities are identities satisfied by the unit and counit of an adjunction.

Statement

Given $C, D$ (categories, or otherwise objects of a $2$-category) with functors (or otherwise morphisms) $L: C \to D$ and $R : D \to C$ and natural isomorphisms (or otherwise $2$-morphisms) $\eta: 1_C \to R \circ L$ and $\epsilon: L \circ R \to 1_D$, the triangle identities are the following:

As equations

$L \stackrel{L\eta}\to L R L\stackrel{\epsilon L}\to L$

and

$R\stackrel{\eta R}\to R L R \stackrel{R\epsilon}\to R$

are identities.

As diagrams

$R\epsilon . \eta R = 1_R$ i.e.

and $\epsilon L . L\eta = 1_L$ i.e.

The RHS of the above diagrams have L and R interchanged. Furthermore, the LHS has C as target of L instead of D.

As string diagrams

In string diagrams, the identities appear as the action of “pulling zigzags straight” (hence the name):

, .

With labels left implicit, this notation becomes very economical: