pushout with zero upper map

Question

If \begin{array}{ccC} A &\xrightarrow {0} & B\\ \downarrow & &\downarrow \\ C & \xrightarrow{g} & D \end{array} is a pushout. is $g$ zero map?

Answer 1

No, for example \begin{array}{ccC} 0 & \to & B\\ \downarrow & &\downarrow \\ C & \xrightarrow{g} & B \oplus C \end{array} is a pushout with $g$ the "inclusion", in say vector spaces.