nLab
cobase change

Cobase change is another term for pushout of a morphism:

if

$$\begin{array}{ccc}X& \to & U\\ \downarrow & & \downarrow \\ Y& \to & Y{\bigsqcup }_{X}U\end{array}$$\array{ X &\to & U \\ \downarrow && \downarrow \\ Y &\to& Y \sqcup_X U }

is a pushout square in some category then the morphism $U\to U{\bigsqcup }_{X}Y$ is the cobase change or cobase extension of $X\to Y$ along $X\to U$.

Remarks

• The dual concept is base change.