Suppose you have two categories $C$ and $D$, and a functor $F: C \to D$. Then we can build a new category as follows:
First, take the union (or coproduct, if you like) of the categories $C$ and $D$.
Then, for each object $X$ in $C$, mapping to $F(X)$ in $D$, add a new morphism from $X \to F(X)$, and extend via composition, so that all the squares formed by composing the new morphisms with the old ones commute.
A good diagram that to make the last requirement clear is this picture from the nCatlab's page on "Functor":
This is supposed to be their graphical depiction of a functor, but it is also kind of a good picture of the category I am talking about. Treat the entire picture as one big category, and then the dotted arrows are the new morphisms, so that all the squares thus generated commute (e.g. the square at $X, Y, F(X),$ and $F(Y)$).
Is there a name for this category? I have done some searching but have not seen a name for this. The best I've thought of so far is that perhaps it could be a very strange version of a slice category in some strange way.