Next we need to show that \$$\tau_1\$$ and \$$\tau_2\$$ factor through this \$$h\$$:

\$\tau_1 = h \triangleright \pi_1\$

\$\tau_2 = h \triangleright \pi_2\$

Now since the category is thin, the above equations must be true. That's because all of the three arrows exist, and they form a well-typed commutative diagram, with domains and codomains of the arrows matching as expected. (Sorry I don't have the bandwidth to draw diagrams now.) Since in a thin category, there's at most one arrow between two objects, the left and right hand sides of these equations must be the same.