i'm having a variance issue that i can't sort out.
1. i believe that the "free cocompletion" functor $PShv : Cat \to Pr^L$ should be a 2-functor, with variance as depicted.
2. the functor $PShv$ is the composite
$$Cat \xrightarrow{(-)^{op}} Cat^{2op} \xrightarrow{Fun(-,Spaces)} (Pr^L)^{2op}~,$$
with variance as depicted. here, $Cat \xrightarrow{Fun(-,Spaces)} Pr^L$ is 1-functorial by left Kan extension, and as for 2-morphisms, a morphism $F \xrightarrow{\alpha} G$ in $Fun(A,B)$ determines a morphism $F_! \to G_!$ in $Fun^L(Fun(A,Spaces),Fun(B,Spaces))$.