What conditions does a functor $F: C\longrightarrow D$ need to satisfy such that its image is a subcategory of $D$?

Question

Is injectivity-on-objects sufficient? Is such a factorization always the epi-mono factorization or are there other subtle differences one needs to be aware of?

Answer 1

Here the image of a functor $F:\mathcal C\to\mathcal D$ is meant in the concrete way:
$\mathrm{im}\, F$ is the collection of objects $F(x)$ for $x\in Ob\,\mathcal C$ and of morphisms $F(f)$ for $f\in Mor\,\mathcal C$.

Now, as the title suggests, this collection might fail to be a subcategory, and this happens only if there are morphisms $f:X\to Y$ and $g:Y'\to Z$ in $\mathcal C$ with $Y\ne Y'$, such that
$$F(Y)=F(Y')\ \ \text{ and }\ \ F(g)\circ F(f)\notin\mathrm{im}\,F\,.$$

In light of this, injectivity on objects is indeed a sufficient condition.

However, regarding the factorization question, the picture is not that clear as for plain algebraic structures.
Exactly because of the above behavior, $\mathrm{im}\,F\,\subseteq\mathcal D$ is not necessarily a subcategory, hence it is not necessarily a category in itself, i.e. not an object of $\mathcal Cat$, so in general, the concrete image cannot be the image with respect to any factorization system on $\mathcal Cat$.