In their book Higher Operads, Higher Categories Tom Leinster claims, that unbiased weak 2-categories may be defined as weak algebras for a strict 2-monad on the strict 2-category of Cat-enriched graphs. The relevant paragraph is:
As in the case of monoidal categories there is an abstract version of the definition of unbiased bicategories phrased in the language of 2-monads. [...] now we take the 2-monad 'free strict 2-category' on Cat-Gph, the strict 2-category of Cat-graphs. [...] The 2-category structure of Cat induces a 2-category structure on Cat-Gph as follows: Given maps $P,Q:B \to B'$ of Cat-graphs, there are only any 2-cells of the form $P\Rightarrow Q$ when $P_0 = Q_0: B_0 \to B_0'$, and in that case a 2-cell $\zeta$ is a family of natural transformations $\zeta_{a,b}: P_{a,b} \to Q_{a,b}$. Now, there is a forgetful map Str-2-Cat$\to$Cat-Gph of 2-categories, [...]
and you can find it on page 93. The book is freely accessible on the arxiv. I do not understand the last sentence. Of course a strict 2-functor does induce a morphism of category enriched graphs. But how does a strict 2-cell $\gamma: F\Rightarrow G, C \to D$ between strict 2-functors induce a 2-cell in Cat-Gph?
A 2-cell of strict 2-functors consist of 1-cells $\gamma_X: FX \to GX$ for each 0-cell $X$ of $C$ such that some equations hold strictly. But nothing forces the object part of $F$ and $G$ to be equal, so how can we define a 2-cell of category enriched graphs between the underlying morphisms of $F$ and $G$? Does Tom Leinster mean icons when they speak about 2-cells in Str-2-Cat?