Is there a name for this category associated to any functor between categories? 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.
 A: It sounds like you're looking for the collage category of the profunctor $F_*:C\not\to D,\ (c,d)\mapsto \hom_D(Fc,d)$ induced by the functor $F$:
There we freely add arrows $c\to Fc$ subject to making the following squares commutative for each arrow $u:c\to c'$:
$$\matrix{c &\overset u\to& c'\\ 
\downarrow &&\downarrow \\ Fc& \underset{Fu}\to &Fc'}$$
A: Consider the category of categories $E$ equipped with functors: \begin{eqnarray*}G_E\colon C\to E,\\H_E\colon D\to E,\end{eqnarray*} and a natural transformation $\eta_E\colon G_E\to H_E\circ F$.
Let morphisms in this category be functors $K\colon E_1\to E_2$ satisfying:
\begin{eqnarray*}KG_{E_1}&=&G_{E_2}&,\\KH_{E_1}&=&H_{E_2}&,\\K(\eta_{E_1})&=&\eta_{E_2}.\end{eqnarray*}
Then your category along with natural inclusions $C\hookrightarrow E,\,D\hookrightarrow E$ is the initial object in this category.
A: This category is also known as a co-comma, which is a type of pushout of the span $D\leftarrow C\to C$. Concretely, it given by taking the weighted colimit with weight $\bullet \to (\bullet \to \bullet)\leftarrow \bullet$, where the first inclusion is at the terminal and the second at the initial object. This gives your construction’s universal property: it represents triples $(f,g,\alpha)$ of a functor $f:C\to E,$ a functor $g:D\to E$, and a natural transformation$ \alpha:f \Rightarrow g\circ F$. It is not precisely a lax pushout, although they’re closely related.
I don’t know of a great literature reference for co-commas among categories, but there are a couple of relevant questions here and on MO:
https://mathoverflow.net/questions/247280/an-explicit-description-of-cocomma-categories
How Co-Comma Categories are constructed?
