Definition 23.4:

Suppose given a pair of morphisms $j_1:C\longrightarrow A_1$ and $j_2:C\longrightarrow A_2$ in a category ${\cal C}$, represented as a diagram:

$\begin{CD} C @> j_1 >> A_1 \\ @V{j_2}VV @. \\ A_2 @. \\ \end{CD}$

a push out is an object $P$ in ${\cal C}$ together with a pair of morphisms $f_1:A_1\longrightarrow P$ and $f_2:A_2\longrightarrow P$ satisfying the following two conditions:

(i) $f_1\circ j_1 = f_2\circ j_2$

(ii) Universal property: Given any pair of morphisms $g_1:A_1\longrightarrow E$ and $g_2:A_2\longrightarrow E$ satisfying

$$g_1\circ j_1 = g_2\circ j_2$$

there exists a unique morphism $\phi:P\longrightarrow E$ such that

$$\phi\circ f_1 = g_1,\quad \phi\circ f_2 = g_2.$$

in