Y such that fg is a morphism. A has a terminate object. Then P is an object whose carrier is a singleton. Next we show that for any object X there is a unique morphism to P . P is a morphism.
P is a morphism, which is obviouse unique as P is a singuleton.
Suppose P' is any object whose carrier is a singuleton. Then the morphism from P' to any terminate object is bijective, therefore an isomorphism. Suppose A is an algebraic category. X of objects factors uniquely as a surjective morphism followed by an injective morphism. X the inclusion.