In category theory:
Assuming there is a category with n objects and j morphisms between them, is it possible that a functor simply adds m new objects and k new morphisms between these, while leaving the original n objects and their morphisms intact?
Would such functor be named an "embedding functor"?
If yes, would it be represented like on this figure? Embedding functor
A functor is a slightly more complicated version of a function. As such it can not modify its domain or codomain. In other words it is just is an assignment between fixed data.
That being said you can do the following. If you are given two categories $\mathcal{C,D}$ you can write them next to each other to obtain a category $\mathcal{C}+\mathcal{D}$, the so called coproduct of $\mathcal{C}$ and $\mathcal{D}$. An object in this category is either an object in $\mathcal{C}$ or an object in $\mathcal{D}$ and the same holds true for morphisms.
Now there is an obvious functor $\operatorname{in}_\mathcal{C}: \mathcal{C}\rightarrow\mathcal{C+D}$ sending an object/morphism in $\mathcal{C}$ to itself considered as object/morphism in $\mathcal{C+D}$.
From the perspective of the category $\mathcal{C}$ the functor $\operatorname{in}_\mathcal{C}$ does indeed freely add the category $\mathcal{D}$ to the category $\mathcal{C}$. But this is a rather unusual way to say that $\operatorname{in}_\mathcal{C}$ is an inclusion of a subcategory.