I'm looking for a reference for a certain pair of monads on $Cat$. One problem is that I don't know the modern way of thinking about some basic things, so excuse me if my presentation is naive.

First some notation. Let $C$ and $D$ be small categories and let $F\colon C\to Cat\;$ and $G\colon D\to Cat\;$ be functors. Let $(F\Uparrow G)$ denote the following category. $$Ob(F\Uparrow G)=\{(c,d,\ell)\;|\;c\in Ob(C), d\in Ob(D), \ell\colon F(c)\to G(d)\}$$ and whose morphisms are ``natural transformation squares", i.e. $$Hom_{(F\Uparrow G)}((c,d,\ell),(c',d',\ell')=\{(f,g,\alpha)\;|\;f\colon c\to c', g\colon d\to d', \alpha\colon G(g)\circ\ell\to\ell'\circ F(f)\}.$$ With the same setup, let $(F\Downarrow G)\;$ denote the category with the same objects but slightly different morphisms, the only difference being that the natural transformation point the other way: $$Ob(F\Downarrow G)=\{(c,d,\ell)\;|\;c\in Ob(C), d\in Ob(D), \ell\colon F(c)\to G(d)\}$$ $$Hom_{(F\Downarrow G)}((c,d,\ell),(c',d',\ell')=\{(f,g,\beta)\;|\;f\colon c\to c', g\colon d\to d', \beta\colon\ell'\circ F(f)\to G(g)\circ\ell\}.$$

Let $D=\{\star\}$, and denote a functor $G\colon D\to Cat\;$ by $\{G\}$. Now let $C=FCat$, some skeleton of the category of finite categories. Then we have a functors $$(FCat\Uparrow\{-\})\colon Cat\to Cat \;\;\;\;\text{ and }\;\;\;\;(FCat\Downarrow\{-\})\colon Cat\to Cat.$$ I think that each is the functor part of some kind of 2-monad on $Cat$. The unit is "constant" and the multiplication is "Grothendieck construction".

Proving that this is associative, etc, looks laborious, and I don't want to reinvent notation, etc. Is there a good reference for these monads, if they really are monads?

Thanks.