Example:

Let us now work in the category Top and recast the gluing lemma in terms of the push-out construction. Take a pair of open sets $G_1, G_2$ in a topological space $X$ and the inclusions

$$j_1:G_1\cap G_2\longrightarrow G_1,\quad j_2: G_1\cap G_2\longrightarrow G_2.$$

The push out for this pair is the space $G_1\cup G_2$ together with inclusion maps

$$i_1:G_1\longrightarrow G_1\cup G_2,\quad i_2:G_2\longrightarrow G_1\cup G_2$$

To see this suppose that $Y$ is a topological space and $f_1:G_1\longrightarrow Y$ and $f_2:G_2\longrightarrow Y$ are a pair of continuous maps such that $f_1\circ j_1 = f_2\circ j_2$ then

$$f_1\Big\vert _{G_1\cap G_2} = f_2\Big\vert _{G_1\cap G_2}$$

The gluing lemma now says that there exists a unique map $\psi:G_1\cup G_2\longrightarrow Y$ such that

$$\psi\Big\vert _{G_1} = f_1, \quad \psi\Big\vert _{G_2} = f_2$$

which means $\psi\circ i_1 = f_1$ and $\psi\circ i_2 = f_2$ as desired. Instead of a pair of open subsets of a topological space one could choose a pair of closed sets.