As a noun pushout is

(category theory) given a pair of morphisms $f:z\backslash rightarrow\; x$ and $g:\; z\backslash rightarrow\; y$ with a common domain, z'', their pushout is a pair of morphisms $i\_1:x\backslash rightarrow\; p$ and $i\_2:y\backslash rightarrow\; p$ as well as their common codomain, ''p'', such that the equation $i\_1\; \backslash circ\; f\; =\; i\_2\; \backslash circ\; g$ is satisfied, and for which there is the ''universal property'' that for any other object ''q for which there are also morphisms $j\_1:\; x\backslash rightarrow\; q$ and $j\_2:\; y\backslash rightarrow\; q$; there is a unique morphism $u:\; p\backslash rightarrow\; q$ such that $u\; \backslash circ\; i\_1\; =\; j\_1$ and $u\; \backslash circ\; i\_2\; =\; j\_2$.