Y are two unitary arrows. Then g is unitary implies that tf is null, and f is unitary implies that t is null. This shows that fg is unitary. If a category has a unitary object then any terminal object is unitary and strict. If a category has a unitary initial then any object is unitary.
T is an arrow.
P is a null arrow, thus with a terminal as codomain. Recall the a zero object in a category is an object which is both an initial and a terminate. We say a category is trivial if any object is a zero object. Suppose the conditions are satisfied. It follows that any object is unitary.