Can it happen that the image of a functor is not a category?

Solution 1:

Consider the category $C$ with four objects, $a,b,c,d$ and, other than identity arrows, a single arrow $a\to b$ and a single arrow $c\to d$. Now consider the category $D$ with three objects $x,y,z$, and, aside from identity arrows, the arrows $x\to y$, $y\to z$, and $x\to z$. Now, consider the functor $F:C\to D$ with $F(a)=x$, $F(b)=F(c)=y$, and $F(d)=z$ (extended uniquely to arrows). Its image is not a category.

This business is related to the fact that epis in $Cat$ are not so simple at all. In work of Isbell epis in $Cat$ are characterized. It's worth noting that regular epis, split epis, etc. in $Cat$ are quite different, attesting again to the subtlety of epis.