The universal monomorphism

Write \(p\colon Y \to \Omega\) for the universal monomorphism, and let \(t\colon * \to \Omega\) be the map classifying the identity \(\id_*\colon * \to *\), exhibited by a pullback square of the form

We claim \(s\) is an inverse to the canonical map \(q\colon Y \to *\), showing \(Y\) is terminal. Since \(q \circ s\) is homotopic to \(\id_*\), it remains to show that \(s \circ q\) is homotopic to \(\id_Y\).To this end, observe that we have two pullback diagrams of the following form:

In particular, both bottom maps \(Y \to \Omega\) are classifying maps for \(\id_Y\), and it follows that \(p\) is homotopic to \(t \circ q\). We then get the chain of homotopies so \(s \circ q\) defines an endomorphism of \(p\) in the slice category \(T_{/\Omega}\). Since \(p\) is a monomorphism, the anima of endomorphisms of \(p\) in \(T_{/\Omega}\) is \((-1)\)-truncated and nonempty (it contains \(\id_Y\)), hence contractible. Therefore \(s \circ q\) is homotopic to \(\id_Y\), as desired.