# When is the image of a Functor a category [duplicate]

Is the essential image of a fully faithful functor always a category? If not is there a counterexample?

The answer in Can it happen that the image of a functor is not a category? does not apply in this situation: the functor there is not full. This question is not asking why the image of a functor is not a category, but it is asking when it is a category.

• But isn't the functor in the example then an equivalence of categories? – sirjoe Nov 13 '19 at 23:56
• I think that example fails to be full as Hom(c,b) is empty but Hom(F(b),F(c)) is non empty since F(b)=F(c)=y – sirjoe Nov 13 '19 at 23:59
• You are right, I judged too quick. Actually, fullness of the functor should be enough. I'll leave my comment still so people can easily find why fullness is really necessary. – Mark Kamsma Nov 14 '19 at 0:05
• Ok, so I have deleted my comment now because it seemed to do more harm than good. The question actually got closed as duplicate, probably because of my comment. So I edited the question to clarify the mistake I made and voted to reopen. – Mark Kamsma Nov 14 '19 at 11:33

If $$F: \mathcal{C} \to \mathcal{D}$$ is full, then the image of $$F$$ is always a category. Clearly we have identity arrows on each object, and composition will be associative since it is associative in $$\mathcal{D}$$. So we only need to check that the image of $$F$$ is closed under composition (see also this answer to see what could go wrong).
Let $$F(f): F(A) \to F(B)$$ and $$F(g): F(C) \to F(D)$$ be arrows in the image of $$F$$, such that $$F(B) = F(C)$$. Then we can compose $$F(f)$$ and $$F(g)$$ in $$\mathcal{D}$$ to get $$F(g)F(f): F(A) \to F(D)$$. Since $$F$$ is full, we find $$h: A \to D$$ such that $$F(h) = F(g)F(f)$$. So $$F(g)F(f)$$ is in the image of $$F$$, and so the image of $$F$$ is closed under composition.
It may also be worth pointing out that if $$F$$ is injective on objects, its image forms a category. Again you need to check that it's closed under composition, which is an easy exercise.