By Barr M.

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The story is diﬀerent for isomorphisms. 8 Proposition Every functor preserves isomorphisms. Proof. This is because the concept of isomorphism is deﬁned in terms of equations involving composition and identity. If f : A − → B is an isomorphism with inverse g, then F (g) is the inverse of F (f ). One of the two calculations necessary to prove this is that F (g) ◦ F (f ) = F (g ◦ f ) = F (idA ) = idF (A) ; the other calculation is analogous. In fact, an analogous proof shows that every functor preserves split monos and split epis.

9). 20 The other two functors which take a set to its powerset are both covariant. The direct or existential image functor takes f : A − → B to the function f∗ : PA − → PB, where f∗ (A0 ) = {f (x) | x ∈ A0 }, the set of values of f on A0 . The universal image functor takes A0 to those values of f which → PB, with → B to f! : PA − come only from A0 : formally, it takes f : A − f! 21 Hom functors Let C be a category with an object C and an arrow f : A − → B. 10, we deﬁned the function Hom(C, f ) : Hom(C, A) − → Hom(C, B) by setting Hom(C, f )(g) = f ◦ g for every g ∈ Hom(C, A), that is for g : C − → A.

The initial category has no objects and, therefore, no arrows. The terminal category has just one object and the identity arrow of that object. To any category C there is just one functor that takes every object to that single object and every arrow to that one arrow. A functor is a monomorphism in Cat if and only if it is injective on both objects and arrows. The corresponding statement for epimorphisms is not true. 6) which takes an object (C, D) to C and an arrow (f, g) : (C, D) − → (C , D ) to f is called the ﬁrst projection.