The coboundary map is defined $\delta^{p} : C^p \to C^{p+1}$, and is the dual of the boundary map. This relation is described by $\delta(\varphi)(c) = \varphi(\partial c)$, where $\varphi \in C^p$ and $c \in C_{p+1}$.
However, supposed we have a $c'$ that's not a boundary, e.g. an edge of a triangle, how is the relation $\varphi(c') = \delta(\varphi)(\color{red}{?})$ defined now?