\documentclass{article}
\input{packages}
\input{def}

\homework{Proofs}

\begin{document}
\maketitle

\begin{exercise}
Let $\mo{trans} : \langle \{a,b,c\}, \{\langle a, b \rangle, \langle b, c \rangle\} \rangle \mto \langle \{a, b, c\}, \{\langle a, b \rangle, \langle b, c \rangle, \langle a, c \rangle\} \rangle$ be a morphism of $\cat{Rel}(2)$ whose underlying function is the identity.
Call its domain $\ob{P}$ and its codomain $\ob{R}$.
Describe, in standard set-theoretic terms, what the pushout of the following is, for any object $\ob{X}$ and morphism $\mo{f} : \ob{P} \mto \ob{X}$:
\begin{center}
\begin{tikzpicture}
\node(p) at (-1,1) {$\ob{P}$};
\node(r) at (1,1) {$\ob{R}$};
\node(x) at (-1,-1) {$\ob{X}$};
\draw[->] (p) -- node[above]{$\mo{trans}$} (r);
\draw[->] (p) -- node[left]{$\mo{f}$} (x);
\end{tikzpicture}
\end{center}
Prove that your description is actually a pushout.
\end{exercise}

\end{document}