A Quick and Simple Dictionary

pushout

Noun.  (category theory) Given a pair of arrows $f:X\; ightarrow\; Y$ and $g:\; X\; ightarrow\; Z$ with a common domain, ''X'', their '''pushout''' is a pair of arrows $u:Y\; ightarrow\; P$ and $v:Z\; ightarrow\; P$ with common codomain, ''P'', such that the equation $ucirc\; f\; =\; vcirc\; g$ is satisfied, and for which there is the ''universal property'' that for any other object ''W'' for which there are also arrows $m:\; Y\; ightarrow\; W$, $n:\; Z\; ightarrow\; W$; there is a unique arrow $k:\; P\; ightarrow\; W$ such that $kcirc\; u\; =\; m$ and $kcirc\; v\; =\; n$.