An "order two sequence" is a sequence of morphisms such that the composition of any two consecutive morphisms is the zero morphism. At the point you are asking about in Waldhausen's paper, the condition is then $\kappa \circ \iota=0$.

References:

[Barry Mitchell, Theory of Categories, Section 15]

[Marco Grandis, On the categorical foundation of homological and homotopical algebra, Section 1.2]

An "order two sequence" is a sequence of morphisms such that the composition of any two consecutive morphisms is the zero morphism. At the point you are asking about in Waldhausen's paper, the condition is then $\kappa \circ \iota=0$ (which always holds by the definition of completed splitting diagram).

References:

[Barry Mitchell, Theory of Categories, Section 15]

[Marco Grandis, On the categorical foundation of homological and homotopical algebra, Section 1.2]

"Order two sequence" seems to mean that the composition of any two consecutive morphisms in the sequence is the zero morphism. At the point you are asking about in Waldhausen's paper, the condition is then $\kappa \circ \iota=0$.

References that are in agreement with the above interpretation:

[Marco Grandis, On the categorical foundation of homological and homotopical algebra, Section 1.2]

[Barry Mitchell, Theory of Categories, Proposition 19.4]

An "order two sequence" is a sequence of morphisms such that the composition of any two consecutive morphisms is the zero morphism. At the point you are asking about in Waldhausen's paper, the condition is then $\kappa \circ \iota=0$.

References:

[Barry Mitchell, Theory of Categories, Section 15]

[Marco Grandis, On the categorical foundation of homological and homotopical algebra, Section 1.2]