TITLE: Can it happen that the image of a functor is not a category?
QUESTION [35 upvotes]: On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a category for the image, and I think they are all satisfied. Am I missing something? Thanks.

REPLY [2 votes]: Drawing a picture of Ittay Weiss's answer:
\begin{align*}
    a \underset{f}{\longrightarrow}\, &b
    \\                &c \underset{g}{\longrightarrow} d
    \\
    \\ &\Big{\downarrow}F
    \\
    \\
    x \underset{F(f)}{\longrightarrow}\,&y \underset{F(g)}{\longrightarrow} z
\end{align*}
Here, the objects of the image are $x$, $y$, and $z$,
and the arrows are $F(f)$ and $F(g)$. Categories are closed under composition of arrows, but $F(g)F(f)$ is not in the image of $F$ (there are no arrows $a \to d$), so the image of $F$ cannot be a category.